GERMS AND MULTIGERMS OF LEGENDRIAN CURVES

M. ZHITOMIRSKII

Abstract. The first main theorem states that in the real analytic category any two diffeomorphic multigerms of Legendrian curves in a fixed contact 3space are contactomorphic. The same holds in the C^{∞} category beyond a certain case of infinite codimension. One of corollaries (and the most difficult part) of this theorem is the equivalence of the following properties of a Legendrian curve multigerm γ: (i) γ does not determine an orientation of ℝ^{3} in the sense that there are contact structures defining different orientations with respect to which γ is Legendrian; (ii) the image of γ admits an orientationreversing symmetry. The second main theorem states the equivalence of (i), (ii), (iii), and (iv), where (iii) is the property that γ is planar, i.e. its image is contained in a non-singular surface, and (iv) is the property that γ is quasihomogeneous. The paper also contains application of these results to contact classification of germs and multigerms of Legendrian curves.

1. Introduction and main results

1.1. Multigerms of Legendrian curves. Reduction theorem. By a multigerm of a curve in ℝ^{3} we mean the collection γ = (γ_{1}, ..., γ_{m}) of germs at 0 ∈ ℝ of maps (ℝ, 0) → (ℝ^{3}, 0) defined up to permutation of the components γ_{i}. A component γ_{i} might be non-singular (γ′_{i}(0) ≠ 0) or singular (γ′_{i}(0) = 0). The image of a multigerm is the union of the images of its components. Two multigerms of curves in ℝ^{3} are called diffeomorphic if their images can be brought one to the other by a local diffeomorphism Φ : (ℝ^{3}, 0) → (ℝ^{3}, 0).

Given a contact 1-form α on ℝ^{3} we denote by (α) the corresponding contact structure - the field of kernels of α. A local contactomorphism of the contact space (ℝ^{3}, (α)) is a local diffeomorphism Φ : (ℝ^{3}, 0) → (ℝ^{3}, 0) preserving the contact structure, i.e. Φ^{∗}α = Qα, where Q is a function germ, Q(0) ≠ 0. Two multigerms of curves in a contact space (ℝ^{3}, (α)) are called contactomorphic if their images can be brought one to the other by a local contactomorphism of this contact space.

A multigerm γ = (γ_{1}, …, γ_{m}) in a contact space (ℝ^{3}, (α)) is called integral or Legendrian if γ_{i}^{∗}α ≡ 0, i = 1, … m.

Theorem 1.1. In the real analytic category any two diffeomorphic Legendrian multigerms in a fixed contact 3-space are contactomorphic. The same holds in the C^{∞} category provided that the multigerms do not belong to a certain set E_{∞} of infinite codimension defined below.

1991 Mathematics Subject Classification. 53D10, 58K50, 14H50.

Key words and phrases. Legendrian curves, multigerms, contactomorphisms.

The work on this paper was supported by the Israeli Science Foundation,

and the Max-Planck-Institut fur Mathematik.

Definition of the set E_{∞}. By E_{∞} in Theorem 1.1 we denote the set of C^{∞} multigerms of curves in ℝ^{3} containing either two non-singular components with infinite order of tangency ^{1} or a singular component whose Taylor series can be reduced by a non-degenerate change of coordinates to a series of the form (f(t), 0, 0).

Remarks.

1. Theorem 1.1 does not hold for Legendrian multigerms of curves in ℝ^{2n+1} if n ≥ 2. In this case there are invariants with respect to the group of contactmorphisms beyond those with respect to the group of diffeomorphisms, see [1] and [16].

2. Theorem 1.1 also holds in the holomorphic category (one has to replace ℝ^{3} by ℂ^{3}). In the holomorphic category for the case of multigerms with one component it was proved by G. Ishikawa in [11] using a general theory of Legendrian maps developed in [12]. The holomorphic category is much simpler than the real analytic category because, as we will explain below, the main difficulty in proving Theorem 1.1 is related to orientation of ℝ^{3} defined by contact structures. (Recall that any contact structure (α) on ℝ^{3} defines an orientation of ℝ^{3} since the sign of the volume form α ∧ dα remains the same when multiplying α by a non-vanishing function). The proof of Theorem 1.1 in the holomorphic category does not require the most difficult part of the proof in the real analytic category, see section 3.5.

The starting point for Theorem 1.1 is a result on structural stability of germs of singular Legendrian submanifolds of a contact manifold (of any odd dimension) proved by A. Givental’ in [10]. This result implies Theorem 1.1 provided that one of the Legendrian multigerms is fixed and the other can be brought to the first one by a local diffeomorphism ϕ with the linear approximation sufficiently close to id.

The possibility to refuse from the assumption that the linear approximation of ϕ is close to id is, from the first look, surprising. In fact, take an orientation-reversing diffeomorphism Ψ and a Legendrian multigerm γ. Assume that the multigerm γ̃ = Ψ ∘ γ is also Legendrian with respect to the same contact structure. Since any contactomorphism preserves orientation of ℝ^{3} then the existence of a contactomorphism Φ bringing the image of γ̃ to the image of γ implies that the image of γ admits an orientation-reversing symmetry (the composition Φ ∘ Ψ). On the other hand, there are Legendrian multigerms (even with one component) whose image does not admit an orientation-reversing symmetry, see examples in Section 2.

1.2. Legendrian multigerms and orientation of the space. The given above counterexample is wrong because the assumption that the image of a Legendrian multigerm γ does not admit an orientation-reversing symmetry cannot hold simultaneously with the assumption that Ψ ∘ γ is Legendrian with respect to the same contact structure for some orientation-reversing diffeomorphism Ψ. In order to explain this in simpler terms we need the following notation and definition that will be used throughout the paper.

1The tangency of two non-singular components γ_{1}, γ_{2} means that the vectors γ′_{1}(0) and γ′_{2}(0)

are proportional. In this case in suitable local coordinates the images of γ_{1} and γ_{2} are described by the equations y = z = 0 and y = g_{1}(x), z = g_{2}(x), where g_{1}(x), g_{2}(x) = o(x). The order of tangency is infinite if the function germs g_{1}(x) and g_{2}(x) have zero Taylor series.

Notation. By Leg(ℝ^{3}) we denote the set of multigerms γ of curves in ℝ^{3} that are Legendrian with respect to some contact structure (α) (depending on γ).

Definition. Let γ ∈ Leg(ℝ^{3}). We will say that γ determines orientation of ℝ^{3} if all contact structures with respect to which γ is Legendrian define the same orientation.

The construction given in the end of Section 1.1 is impossible because one has the following statement:

(*) Assume that a multigerm γ ∈ Leg(ℝ^{3}) does not determine orientation of ℝ^{3}. Then in the real analytic category the image of γ admits an orientation-reversing symmetry. If γ ∉ E_{∞} then the same holds in the C^{∞} category.

This statement is a corollary of Theorem 1.1. On the other hand, (*) is the most difficult part of this Theorem, modulo well-developed techniques.

1.3. Four equivalent properties of Legendrian multigerms. Theorem 1.2 below states the equivalence of four properties of any multigerm γ ∈ Leg(ℝ^{3}) - the two properties given in (*), the planarity, and the quasi-homogeneity.

Definition. A multigerm γ of a curve in ℝ^{3} is planar if its image belongs to a non-singular surface (2-dimensional submanifold) of ℝ^{3}.

Definition. A multigerm of a curve in ℝ^{3} is called quasi-homogeneous if there exist λ_{1}, λ_{2}, λ_{3} > 0 (weights) such that γ is RL-equivalent ^{2} to a multigerm whose components have the form

x = at^{rλ_{1}}, y = bt^{rλ_{2}}, z = ct^{rλ_{3}}, a, b, c, r ∈ ℝ,

where the local coordinate system (x, y, z) and the weights λ_{1}, λ_{2}, λ_{3} > 0 are the same for all components and the numbers a, b, c, r depend on a component.

Theorem 1.2. In the real analytic category (respectively the C^{∞} category) the following statements are equivalent for any multigerm γ ∈ Leg(ℝ^{3}) (respectively for any multigerm γ ∈ Leg(ℝ^{3}) beyond the set E_{∞}):

(i) γ does not determine orientation of ℝ^{3}; (ii) the image of γ admits an orientation-reversing symmetry; (iii) γ is planar; (iv) γ is quasi-homogeneous.

The implications (iii) ⟹ (ii) ⟹ (i) are trivial, but the other implications are not. Note that each of the implications (ii) ⟹ (iii), (iii) ⟹ (ii), (iii) ⟹ (iv) and (iv) ⟹ (iii) is violated by certain multigerms (even with one component) of space curves. By Theorem 1.2 such multigerms do not belong to Leg(ℝ^{3}), i.e. they are not Legendrian with respect to any contact structure on ℝ^{3}.

2Two multigerms γ = (γ_{1}, ..., γ_{m}) and γ̃ = (γ̃_{1}, ..., γ̃_{m}) are called RL-equivalent if they can be brought one to the other by a local diffeomorphism Φ : (ℝ^{3}, 0) → (ℝ^{3}, 0) (change of coordinates)

and local diffeomorphisms ϕ_{i} : (ℝ, 0) → (ℝ, 0), i = 1, ..., m (independent reparameterization of the components), i.e up to permutation of the components one has γ̃_{i} = Φ ∘ γ_{i} ∘ ϕ_{i}, i = 1, ..., m.

Theorem 1.3. (Corollary of Theorem 1.2). In the real analytic category (respectively the C^{∞} category) any multigerm γ ∈ Leg(ℝ^{3}) (respectively any multigerm γ ∈ Leg(ℝ^{3}) beyond the set E_{∞}) satisfies one of the following two conditions:

(a) γ is planar and quasi-homogeneous;

(b) γ is neither planar nor quasi-homogeneous.

It is clear that any planar quasi-homogeneous multigerm belongs to Leg(ℝ^{3}). Within multigerms that are neither planar nor quasi-homogeneous there are multigerms both in and beyond the class Leg(ℝ^{3}), see Example 2.5 and Sections 6-8.

1.4. Plan for the paper. In Section 2 we give several simple examples illustrating Theorems 1.2, 1.3 and the notion of the determinacy of orientation by a Legendrian multigerm. Theorem 1.1 is proved in Section 3 using the well known techniques (the homotopy method) and the most difficult implication (i) ⟹ (ii) in Theorem 1.2. The proof of Theorem 1.2 is rather long - it is contained in Sections 4-5.

In sections 6-10 we present applications of Theorems 1.1 and 1.2 to contact classification of Legendrian multigerms in a fixed contact space. ^{3}

The simplest application holds for multigerms of classes A_{i}, section 6. The image of multigerms of the class A_{i} is diffeomorphic to the singular submanifold of ℝ^{3} given by the equation y^{2} = x^{i+1} = z = 0. The class A_{2}∪A_{4}∪⋯ consists of singular curves with non-zero 2-jet whose Taylor series in not RL-equivalent to (t^{2}, 0, 0). The class A_{1}∪A_{3} ∪ ⋯ consists of multigerms with two components; the components are non-singular curves with a finite order of tangency at the origin. It is easy to see that A_{i} ∩ Leg(ℝ^{3}) =A_{i}, i.e. any multigerm of the class A_{i} is Legendrian with respect to some contact structure. This makes application of Theorem 1.1 very simple - by this theorem all Legendrian multigerms of the class A_{i} in a fixed contact space are contactomorphic.

Section 7 is devoted to contact classification of E singularities - germs of singular curves with zero 2-jet, non-zero 3-jet, and the Taylor series not RL-equivalent to (t^{3}, 0, 0). The classification of such singularities with respect to diffeomorphisms is obtained in [9], see also [2]. It is described, in the notations in [2], by normal forms E_{6k,p,i} and E_{6k+2,p,i}, where k ≥ 1, 0 ≤ p ≤ k, 0 ≤ i ≤ min(k − 1, p). We prove that E_{6k,p,i}, E_{6k+2,p,i} ∈ Leg(ℝ^{3}) if and only if p = i + 1. Within the class E ∩ Leg(ℝ^{3}) there are planar quasi-homogeneous multigerms (the case p = k, i = k − 1) and multigerms that are neither planar nor quasi-homogeneous (the case p = i+1 < k). Theorem 1.1 implies that in a fixed contact space all multigerms diffeomorphic to E_{6k,i+1,i} (respectively E_{6k+2,i+1,i}) are contactomorphic.

In Section 8 we obtain few more classification results for germs of Legendrian curves (multigerms with one component); this allows us to determine and classify all contact-simple germs of Legendrian curves. ^{4} We prove that a germ γ of an Legendrian curve is contact-simple within Legendrian germs (i.e. any Legendrian germ sufficiently close to γ is contactomorphic to one of a finite number of fixed Legendrian germs) if and only if γ is simple as a space curve, with respect to the group of diffeomorphisms (i.e. any space curve germ sufficiently close to γ is diffeomorphic to one of a finite number of fixed space curve germs). Therefore our theorem on determination of contact-simple germs of Legendrian curves is the same as the theorem in [9] on determination of simple germs of space curves. Nevertheless, the classification of contact-simple germs of Legendrian curves differs from the classification of simple space curves in [9] because there are many simple singularities of space curves that are not Legendrian with respect to any contact structure, in particular quasi-homogeneous non-planar curves and planar non-quasi-homogeneous curves which do not belong to Leg(ℝ^{3}) by Theorem 1.3. On the other hand, we prove that there is a 1-1 correspondence between contact-simple singularities of Legendrian curves in ℝ^{3} and simple singularities of curves in ℝ^{2}, determined and classified in [8]. This correspondence is not canonical - it holds in a local coordinate system in which the contact structure is described by the 1-form dz − ydx and requires a (non-canonical) projection (x, y, z) → (x, y).

3Theorems 1.1 and 1.2 also can be applied for classification of germs of Legendrian curves in a 4-dimensional Engel manifold and its generalization - an n-dimensional ”monster” manifold constructed in [14] using expressed in modern terms E. Cartan prolongation procedure. We hope that this application also leads to a number of results relating singularities of plane and space curves with singularities of Cartan-Goursat flags, see [14]. We also hope that there are applications in the theory of implicit ODE’s and in the theory of Legendrian knots.

4Results of Section 8 were obtained in the author’s work [17], by a different method, using certain pre-normal form for Legendrian curves.

In section 9 we classify, with respect to contactomorphisms, Legendrian multigerms (γ_{1}, γ_{2}) such that γ_{1} is a curve with a cusp singularity (a curve diffeomorphic to (t^{2}, t^{2k+1}, 0)) and γ_{2} is a non-singular curve. We denote this class by (A_{2k}, l). We use the classification of multigerms of this class with respect to the group of diffeomorphism obtained in [13]. At first we consider the case k = 1. Using Theorem 1.2 we prove that a multigerm in (A_{2}, l) is Legendrian with respect to some contact structure if and only if it is is planar and quasi-homogeneous (the class (A_{2}, l) has singularities violating these properties). Theorem 1.1 implies that in a fixed contact space any Legendrian multigerm (γ_{1}, γ_{2}) ∈ (A_{2}, l) is contactomorphic to one of two normal forms without parameters. These two normal forms correspond to the cases that the non-singular component γ_{2} is not tangent (respectively tangent) to the limit tangent line l(γ_{1}) to the image of the component γ_{1}.

The contact classification of multigerms of the class (A_{2k}, l) is a bit more involved if k ≥ 2. In this case (A_{2k}, l), but for k ≥ 2 the class (A_{2k}, l) ∩ Leg(ℝ^{3}) contains certain singularities that are neither planar nor quasi-homogeneous. The class (A_{2k}, l) ∩ Leg(ℝ^{3}) can be completely described using Theorem 1.2. Using Theorem 1.1 we prove that in a fixed contact space any multigerm (γ_{1}, γ_{2}) ∈ (A_{2k}, l) is contactomorphic to one and only one of 2k normal forms without parameters.

In Section 10 we use Theorems 1.1 - 1.3 in order to obtain a contact classification of Legendrian multigerms with three non-singular components.

In the Appendix we discuss an alternative way of contact classification of Legendrian multigerms in the fixed contact space (ℝ^{3}, (dz − ydx)), using the projections π_{x,y} : (x, y, z) → (x, y) and π_{x,z} : (x, y, z) → (x, z). Let γ, γ̃ be Legendrian multigerms. We prove that γ and γ̃ are contactomorphic if one of the following holds: (a) the projections π_{x,z}γ and π_{x,z}γ̃ are diffeomorphic (via any local diffeomorphism) provided that these projections do not contain components with the zero Taylor series; (b) the projections π_{x,y}γ̃ are diffeomorphic via a local diffeomorphism preserving the volume form dx ∧ dy up to multiplication by a number.

In view of these sufficient conditions and results in [17] and Section 10 we give example showing that in general the contact classification of germs and multigerms of Legendrian curves in ℝ^{3} cannot be reduced to a classification of plane curves. Namely, we give examples illustrating the following statements:

1. If two Legendrian multigerms γ and γ̃ in the contact space (ℝ^{3}, (dz − ydx)) are contactomorphic then (a) the projections π_{x,z}γ and π_{x,z}γ̃ do not need to be diffeomorphic; (b) the projections π_{x,y}γ and π_{x,y}γ̃ do not need to be diffeomorphic;

2. If two Legendrian multigerms γ and γ̃ in the contact space (ℝ^{3}, (dz − ydx)) have diffeomorphic projections π_{x,y}γ and π_{x,y}γ̃ then γ and γ̃ do not need to be diffeomorphic (and consequently contactomorphic).

Examples illustrating these statements for the projection π_{x,z} are very simple, they hold already for E singularities. In the case of the projection π_{x,y} the singularities in the examples are deeply degenerate. This is in correspondence with results in [17] and Section 10.

Acknowledgements. I am thankful to G. Ishikawa and R. Montgomery for a number of stimulating questions and comments during my work in the direction of Theorems 1.1 - 1.3 and their applications.

2. Examples.

The following examples illustrate Theorems 1.2 and 1.3 and the notion of determinacy of orientation of the space by a Legendrain multigerm. Example 2.1. Consider a planar quasi-homogeneous multigerm of the form

γ = (γ_{1}, ..., γ_{m}), γ_{i} : x = a_{i}t^{q}, y = b_{i}t^{p}, z = 0,

where a_{i}, b_{i} ∈ ℝ. The multigerm γ is Legendrain with respect to any contact structure of the form (dz + λ_{1}ydx + λ_{2}xdy) with λ_{1}, λ_{2} satisfying the relation qλ_{1} + pλ_{2} = 0. Within such contact structures there are those that define different orientations of ℝ^{3}, for example (dz + pydx − qxdy) and (dz − pydx + qxdy).

Example 2.2. Consider the multigerm γ = (γ_{1}, γ_{2}) with two components

γ_{1} : x = t^{2}, y = t^{5}, z = 0, γ_{2} : x = t, y = t^{2}, z = t^{3}.

It is Legendrain with respect to the contact structure dz + λ_{1}ydx + λ_{2}xdy with λ_{1} = −15, λ_{2} = 6. It is easy to check that γ is not planar. By Theorem 1.2 the multigerm γ determines orientation of ℝ^{3}, the image of γ does not admit an orientation-reversing symmetry, and γ is not quasi-homogeneous.

Example 2.3. Let us show, using Theorems 1.2 and 1.3, that the multigerm

γ = (γ_{1}, γ_{2}), γ_{1} : x = t^{2}, y = t^{5}, z = 0, γ_{2} : x = t, y = t^{3}, z = at^{2}

is not Legendrain with respect to any contact structure on ℝ^{3} for any a ∈ ℝ.

If a ≠ 0 then γ admits an orientation-reversing symmetry, for example (x, y, z) → (x, −y + \frac{2xz}{a} , z). On the other hand, if a ≠ 0 then it can be easily showed that γ is not planar. Therefore γ ∉ Leg(ℝ^{3}) by Theorem 1.2. In the case a = 0 γ ∉ Leg(ℝ^{3}) by a different part of Theorem 1.2, namely, by Theorem 1.3: in this case γ is planar, but one can prove that it is not quasi-homogeneous.

Example 2.4. Consider the following germ of a singular curve:

ψ : x = t^{3}, y = t^{7} + at^{8}, z = bt^{11}, a, b ∈ ℝ.

This germ has the following properties:

(a) It is easy to check that if b ≠ 0 then ψ is not a planar curve (for any a ∈ ℝ);

(b) If a ≠ 0 and b ≠ 0 then ψ ∈ Leg(ℝ^{3}). For example, ψ is Legendrain with respect to the contact structure (dz + λ_{1}ydx + λ_{2}xdy), where (λ_{1}, λ_{2}) is the solution of the linear system 3λ_{1} + 7λ_{2} = 0, 3λ_{1} + 8aλ_{2} + 11b = 0 (λ_{1} == \frac{77b}{3a}, λ_{2} = −\frac{11b}{a}; )

(c) If a = 0 then the image of ψ admits an orientation-reversing symmetry, for example (x, y, z) → (−x, −y, −z);

(d) If a ≠ 0 then it is easy to prove that the germ ψ is not quasi-homogeneous. These properties and Theorems 1.2 and 1.3 imply the following statements:

(i) If a ≠ 0 and b ≠ 0 then the curve ψ belongs to Leg(ℝ^{3}) and defines orientation of ℝ^{3}. The image of the curve ψ does not admit an orientation-reversing symmetry; (ii) If a = 0, b ≠ 0 or a ≠ 0, b = 0 then ψ ∉ Leg(ℝ^{3}) (if a = 0 and b ≠ 0 then the curve ψ admits an orientation-reversing symmetry, but it is not planar; if a ≠ 0 and b = 0 then ψ is planar, but not quasi-homogeneous).

Example 2.5. Many singularities of space curves are described by the normal form

ψ : x = t^{q}, y = t^{p} ± t^{s}, z = t^{r}, q < p < s < r  (2.1)

where the integers q, p, s, r satisfy the following assumptions:

(a) p is not divisible over q; (b) s ∉ {α_{1}q + α_{2}p}, α_{1}, α_{2} are integers ≥ 0;

(c) s + q is not divisible over p; (d) r ∉ {α_{1}q + α_{2}p}, α_{1}, α_{2} are integers ≥ 0. The explanation of (a) - (d) is as follows, see [8], [9]. If one of the assumptions

(a)-(c) is violated then the germ ψ is ψ is quasi-homogeneous - it is RL-equivalent to (t^{q}, t^{p}, t^{r}) or (t^{q}, t^{s}, t^{r}). One can prove that under assumptions (a) - (c) the germ ψ is not quasi-homogeneous. If (d) is violated then the r-jet of ψ can be simplified to (t^{q}, t^{p} ± t^{s}, 0). If q = 3 then under assumptions (a)-(d) ψ is not planar, see [9]. This is not so if q ≥ 4. For example the germ (t^{4}, t^{6} + t^{7}, t^{9+2k}) satisfies (a)-(d) for any k ≥ 0, but if k ≥ 4 then it is RL-equivalent to (t^{4}, t^{6} + t^{7}, 0), see [9].)

Proposition 2.1. Let ψ be a germ of form (2.1) with q, p, s, r satisfying (a)-(d). Then ψ ∈ Leg(ℝ^{3}) if and only if r = s + q.

Proof. Assume that ψ is Legendrain with respect to a contact 1-form α = Adx + Bdy + Cdz. Assumptions (a) and (b) easily imply that A(0) = B(0) = 0. Consequently C(0) ≠ 0 and there is no loss of generality to assume that C ≡ 1. Then the functions A and B satisfy the relation

F (t) = A(t^{q}, t^{p} ± t^{s}, t^{r}) · q · t^{q−1} + B(t^{q}, t^{p} ± t^{s}, t^{r}) · (pt^{p−1} ± st^{s−1}) + r · t^{r−1} ≡ 0.

Let f_{1}t + f_{2}t^{2} + ⋯ be the Taylor series of F (t). Using (a)-(d) it is easy to obtain

f_{p+q−1} = q · \frac{∂A}{∂y}(0) + p · \frac{∂B}{∂x} (0) = 0; f_{s+q−1} = ± (q · \frac{∂A}{∂y}(0) + s · \frac{∂B}{∂x} (0)) = δ,

where δ = 0 if r ≠ q + s and δ = r if r = q + s. It follows that if r ≠ q + s then (∂A/∂y)(0) = (∂B/∂x)(0) = 0. In this case (α ∧ dα)(0) = 0, i.e. the 1-form α is not contact. The contradiction means that if r ≠ q + s then ψ ∉ Leg(ℝ^{3}). If r = q + s then ψ ∈ Leg(ℝ^{3}) because ψ is Legendrain with respect to the contact 1-form dz + λ_{1}ydx + λ_{2}xdy with (λ_{1}, λ_{2}) being the solution of the linear system qλ_{1} + pλ_{2} = 0, qλ_{1} + sλ_{2} = ±r. □

Remind that under conditions (a)-(d) the germ (2.1) is not quasi-homogeneous. Therefore if r = s + q then by Theorems 1.2, 1.3 and 2.1 the germ (2.1) is not planar and determines orientation of ℝ^{3}. The latter can also be seen from the proof of Proposition 2.1: if ψ is Legendrain with respect to a contact 1-form α = dz + Adx + Bdy then (α ∧ dα)(0) = (λ_{2} − λ_{1})dx ∧ dy ∧ dz, where λ_{1} and λ_{2} are uniquely determined by the numbers q, p, s, r.

3. Proof of Theorem 1.1 (using Theorem 1.2)

In this section we prove Theorems 1.1 using the implication (i) ⟹ (ii) in Theorem 1.2. We work in a fixed category which is either real analytic or C^{∞}. At the end of the section we also prove Theorem 1.1 in the holomorphic category.

3.1. Reformulation of Theorem 1.1. At first we reformulate Theorem 1.1.

Theorem 3.1. (equivalent to Theorem 1.1). Let γ be a multigerm of a curve in ℝ^{3}. In the C^{∞} category assume that γ ∉ E_{∞}. If γ is Legendrain with respect to contact structures (α) and (β) then there exists a local diffeomorphism preserving the image of γ and sending (β) to (α).

Theorem 1.1 implies Theorem 3.1 by the Darboux theorem on the diffeomorphness of all local contact structures. Let us show that Theorem 3.1 implies Theorem 1.1. Let γ and γ̃ be Legendrain multigerms with respect to a contact structure (α) in Theorem 1.1, and let Φ be a local diffeomorphism sending the image of γ̃ to the image of γ. Let γ̂ = Φ ∘ γ̃. The multigerms γ and γ̂ have the same image and consequently in the real analytic category γ̂ is Legendrian with respect to (α). The same holds in the C^{∞} category due to the assumption γ, γ̃ ∉ E_{∞}. ^{5} On the other hand γ̂ is Legendrain with respect to the contact structure (β) = (Φ^{∗}α). By Theorem 3.1 there exists a local diffeomporphism Ψ sending (β) to (α) preserving the image of γ̂. Then the composition Ψ ∘ Φ is a local contactomorphism of the contact space (ℝ^{3}, (α)) sending the image of γ̃ to the image of γ.

3.2. The case that the contact structures (α) and (β) are transversal. The transversality of (α) and (β) means that (α ∧ β)(0) ≠ 0. This is an easy case in Theorem 3.1. In this case the vector field Y defined by the relation Y ⌋ Ω = α ∧ β, where Ω is a non-degenerate volume form on ℝ^{3}, does not vanish at 0 and the image of the multigerm γ belongs to the phase curve l of Y containing 0. This phase curve l is a non-singular 1-dimensional submanifold of ℝ^{3} and it is Legendrain with respect to the contact structures (α) and (β). By Darboux-Givental’ theorem (see [4]) there exists a local diffeomorphism sending (β) to (α) and preserving l pointwise. Consequently, this diffeomorphism preserves the image of γ (also pointwise).

5in order to conclude that γ̂^{∗}α ≡ 0 it suffices to assume that the multigems γ and γ̂ have no components with the zero Taylor series and consequently any component is immersed at any point t ≠ 0 sufficiently close to 0.

3.3. The case that the contact structures (α) and (β) are tangent. Reduction to the case that they define the same orientation. We have reduced Theorem 3.1 to the case that the 1-forms α and β are tangent at 0: (α ∧ β)(0) = 0. Since α and β are non-vanishing 1-forms we may assume that α(0) = β(0).

Now we use the implicatioin (i) ⟹ (ii) in Theorem 1.2. It allows to assume, without loss of generality, that the contact structures (α) and (β) in Theorem 3.1 define the same orientation of ℝ^{3}. In fact, assuming that we have proved Theorem 3.1 for this case, the proof for the case that (α) and (β) define different orientation is as follows. By the statement (i) ⟹ (ii) in Theorem 1.2 the image of γ admits an orientation-reversing symmetry Φ. It follows that γ is Legendrain with respect the contact structures (Φ^{∗}α) and (β). These contact structures define the same orientation of ℝ^{3} and therefore there exists a local diffeomorphism Ψ sending (β) to (ϕ^{∗}α) preserving the image of γ. The composition Φ^{−1} ∘ Ψ preserves the image of γ and sends (β) to (α).

We have reduced Theorem 3.1 to the following proposition.

Proposition 3.2. Let γ be a multigerm of a curve in ℝ^{3}. In the C^{∞} category assume that γ ∉ E_{∞}. Assume that γ is Legendrain with respect to germs of contact 1-forms α and β satisfying the following assumptions:

(a) α(0) = β(0) = 0;

(b) the orientations on T_{0}ℝ^{3} defined by α and β are the same. Then there exists a local diffeomorphism Φ which preserves the image of γ and brings (β) to (α).

3.4. Proof of Proposition 3.2. The assumptions (a) and (b) allow to prove this proposition by the homotopy method, see [6] and [18]. Let α_{s} = α+s(β−α). We will prove that that there exists a family of local diffeomorphism Φ_{s}, s ∈ [0, 1] preserving the image of the multigerm γ and such that Φ_{s}^{∗}(α_{s}) = α_{0} for any s ∈ [0, 1].

Lemma 3.3. Under assumptions (a) and (b) the germ α_{s} = α+s(β −α) is contact for all s ∈ [0, 1], i.e. (α_{s} ∧ dα_{s})(0) ≠ 0.

Proof. Fix a non-degenerate volume form Ω on T_{0}ℝ^{3} and consider the function F (s) such that (α_{s} ∧ dα_{s})(0) = F (s)Ω. Condition (a) implies that F (s) is a linear function: F (s) = k_{1}s + k_{2} for some k_{1}, k_{2} ∈ ℝ. Condition (b) implies that F (0) and F (1) are non-zero numbers of the same sign. Therefore F (s) ≠ 0 for any s ∈ [0, 1]. □

Lemma 3.3 allows to define a family of germs of vector fields Y_{s} by the relation

Y_{s} ⌋ (α_{s} ∧ dα_{s}) = α ∧ β, s ∈ [0, 1].  (3.1)

By condition (a) one has Y_{s}(0) = 0. This allows to define a family Φ_{s}, s ∈ [0, 1], of local diffeomorphisms by the system of ODEs

\frac{dΦ_{s}}{ds} = Y_{s}(Φ_{s}), Φ_{0} = id.  (3.2)

We will show that Φ_{s} is the required family of local diffeomorphisms: Φ_{s} preserves the image of the multigerm γ and sends the contact structure (α_{s}) to the (α_{0}).

Take a point p ≠ 0 close to 0 ∈ ℝ^{3} and belonging to the image of γ. Let γ_{i} be a component of γ passing through p: p = γ_{i}(t_{0}), t_{0} ≠ 0. In the real analytic category γ′_{i}(t_{0}) ≠ 0. The same holds in the C^{∞} category due to the assumption γ ∉ E_{∞}.

Therefore in a small enough neighborhood of p the image of the the component γ_{i} is a non-singular 1-dimensional submanifold l_{p} ⊂ ℝ^{3}. Since γ is Legendrain with respect to 1-forms α and β then l_{p} is tangent to the 2-spaces kerα(p) and kerβ(p). If (α∧β)(p) ≠ 0 then (3.1) implies that Y_{s}(p) ∈ kerα(p)∩kerβ(p). if (α∧β)(p) = 0 then by (3.1) one has Y_{s}(p) = 0. Consequently either Y_{s}(p) = 0 or the vector Y_{s}(p) is tangent to the image of γ_{i}. It follows that the family of local diffeomorphisms defined by (3.2) preserves the image of γ_{i} for any s ∈ [0, 1]. Therefore Φ_{s} preserves the image of the multigerm γ.

Now we will prove that Φ_{s}^{∗}(α_{s}) = α_{0}, i.e that there exists a family H_{s} of function germs such that H_{s}(0) ≠ 0 and H_{s} · Φ^{∗}α_{s} = α_{0}. Note that we have proved above that Y_{s} ⌋ α = Y_{s} ⌋ β = 0 for any s ∈ [0, 1]. Consequently Y_{s} ⌋ α_{s} = 0. Let L_{Y_{s}} be the Lie derivative along the vector field Y_{s}. Then L_{Y_{s}}α_{s} = Y_{s} ⌋ dα_{s}. This relations, (3.1), and the relation β = \frac{dαs}{ds} imply

(L_{Y_{s}}α_{s} + \frac{dα_{s}}{ds} ) ∧ α_{s} = 0.

Since α_{s}(0) ≠ 0 for any s ∈ [0, 1] it follows that there exists a family h_{s} of function germs, s ∈ [0, 1], such that

L_{Y_{s}}α_{s} + h_{s}α_{s} + \frac{dα_{s}}{ds} = 0.  (3.3)

Define a family H_{s} of function germs by the ODE

\frac{dH_{s}}{ds} = h_{s}(Φ_{s}) · H_{s}, H_{0} ≡ 1.  (3.4)

Consider the function G(s) = H_{s} · Φ_{s}^{∗}α_{s} whose values are 1-forms. Differentiating this function using (3.2) and (3.4) we obtain

G′(s) = H_{s} · Φ_{s}^{∗} [L_{Y_{s}}α_{s} + h_{s}α_{s} + \frac{dα_{s}}{ds} ] .

By (3.3) G′(s) ≡ 0. Since G(0) = α_{0} by the initial conditions in (3.2) and (3.4) then G(s) ≡ α_{0}, i.e H_{s} · Φ_{s}^{∗}α_{s} ≡ α_{0}. It remains to note that H_{s}(0) ≠ 0 because the ODE (3.4) is linear.

3.5. Proof of Theorem 1.1 in the holomorphic category. A little modification of the given arguments gives a complete proof of Theorem 1.1 in the holomorphic category. In the holomorphic category Theorem 1.1 also can be reformulated as Theorem 3.1 and the reduction to the case α(0) = β(0) is the same. The rest of the proof is the same if the path α_{s} = α + s(β − α), s ∈ [0, 1] consists of contact 1-forms. If this is not so then the path α_{s} should be replaced by another path of the form α̃_{s} = α + f(s)(β − α), where f(s) : [0, 1] → ℂ is a smooth function such that f(0) = 0, f(1) = 1 and (α_{s} ∧ dα_{s})(0) ≠ 0 for all s ∈ [0, 1]. It is clear that such a complex-valued function f(s) exists. The family of vector fields Y_{s} defined by (3.1) should be replaced by the family Ỹ_{s} such that

Ỹ_{s} ⌋ (α̃_{s} ∧ dα̃_{s}) = α̃_{s} ∧ \frac{dα̃_{s}}{ds} = f′(s) · α ∧ β.

The given above arguments with minor modifications imply that the family of local diffeomorphisms Φ_{s} defined by (3.2) with Y_{s} replaced by Ỹ_{s} preserves the image of γ and brings the contact structures (α̃_{s}) to (α̃_{0}) = (α).

4. Proof of Theorem 1.2

4.1. Outline of the proof. As we mentioned in Section 1, the implications (iii) ⟹ (ii) ⟹ (i) in Theorem 1.2 are trivial. In order to prove Theorem 1.2, we have to prove the implications (iv) ⟹ (i) and (i) ⟹ (iii), (iv).

The implication (iv) ⟹ (i) is proved in Section 4.2. The implications (i) ⟹ (iii), (iv) can be joined into the following statement.

Theorem 4.1. Let γ be a multigerm of a curve in ℝ^{3}. In the C^{∞} category assume that γ ∉ E_{∞}. If γ is Legendrain with respect to contact structures (α) and (β) defining different orientations of ℝ^{3} then γ is a planar and quasi-homogeneous.

In what follows we will prove this theorem under the following assumption:

(A) the image of γ is not contained in the union l_{1} ∪ l_{2} of two non-singular nontangent 1-dimensional submanifolds of ℝ^{3}.

Let us show that there is no loss of generality to assume (A) - if (A) is violated then the conclusion Theorem 4.1 is trivial. In fact, take local coordinates in which l_{1} and l_{2} are the x- and y-axes. The image of any component of γ is contained either in l_{1} or in l_{2}, therefore in the real analytic category each of the components has, up to reparameterization, the form (at^{p}, 0, 0) or (0, bt^{q}, 0). In the C^{∞} category the assumption γ ∉ E_{∞} implies that each of the component of γ has, up to reparameterization, the form (t, 0, 0) or (0, t, 0). In any case the multigerm γ is planar and quasi-homogeneous with weights (1, 1, 1).

Introduce the vector field Y_{α,β} defined by the relation

Y_{α,β} ⌋ Ω = α ∧ β,  (4.1)

where Ω is a non-degenerate volume form on ℝ^{3}. We will say that Y_{α,β} is a vector field associated to the contact structures (α) and (β). It is defined up to multiplication by a non-vanishing functions since the 1-forms α and β and the volume form Ω are defined up to multiplication by a non-vanishing function.

Definition. A germ Y of a vector field on ℝ^{3} is tangent to a germ ψ : (ℝ, 0) → (ℝ^{3}, 0) of a curve in ℝ^{3} if the vectors ψ̇(t) and Y (ψ(t)) are linearly dependent for any t close to 0. The vector field Y is tangent to a multigerm of a curve in ℝ^{3} if it is tangent to each of the components of the multigerm.

The integrability of the multigerm γ with respect to the contact structures (α) and (β) implies that the vector field Y_{α,β} is tangent to γ. In fact, take a point p ∈ ℝ^{3} near the origin and let γ_{i} be a component of γ whose image contains p: γ_{i}(t_{0}) = p. If the contact structures (α) and (β) are transversal at p then the vectors γ′(t_{0}) and Y_{α,β}(p) are contained in the same line kerα(p) ∩ kerβ(p). If the contact planes kerα(p) and kerβ(p) coincide then by (4.1) Y_{α,β}(p) = 0.

Assumption (A) implies that the contact forms α and β in Theorem 4.1 are tangent at 0: (α ∧ β)(0) = 0, see Section 3.2. Consequently Y_{α,β}(0) = 0. The proof of Theorem 4.1 is based on the analysis of the singularity of the vector field Y_{α,β}.

Proposition 4.2. Let α and β be the 1-forms in Theorem 4.1 and let (α∧β)(0) = 0. Within the case (A) the vector field Y_{α,β} has the following properties:

(i) the germ of set of singular points of Y_{α,β} is a non-singular 1-dimensional submanifold of ℝ^{3}; (ii) the eigenvalues of the vector field Y_{α,β} at the origin are λ_{1}, λ_{2}, 0, where λ_{1}, λ_{2} are non-zero real numbers of the same sign.

We prove Proposition 4.2 in a separate section 5. This proposition is the most difficult part of the proof of Theorem 4.1.

Properties (i) and (ii) in Proposition 4.2 allow to bring Y_{α,β}, by a change of coordinates, to the following normal form.

Proposition 4.3. (real analytic and C^{∞} categories). Any germ of a vector field satisfying (i) and (ii) in Proposition 4.2 is diffeomorphic to a vector field

ẋ = f_{1}(x, y, z), ẏ = f_{2}(x, y, z), ż = 0,  (4.2)
f_{1}(0, 0, z) = f_{2}(0, 0, z) ≡ 0.  (4.3)

This proposition is proved in Section 4.3. Propositions 4.2 and 4.3 allow to fix local coordinates in which the vector field Y_{α,β} has form (4.2)-(4.3).

Proposition 4.4. Let γ be a multigerm of a curve in ℝ^{3} which is tangent to a vector field of the form (4.2)-(4.3). If γ satisfies (A) and in the C^{∞} category γ ∉ E_{∞} then the image of γ is contained in the plane z = 0.

Proposition 4.4 is proved in Section 4.4. Propositions 4.2, 4.3 and 4.4 imply that the multigerm γ in Theorem 4.1 is planar. In order to prove that γ is quasihomogeneous we consider the restriction Y of the vector field Y_{α,β} to the plane z = 0. It is a vector field on ℝ^{2}(x, y) with non-zero real eigenvalues of the same sign at the singular point 0 ∈ ℝ^{2}. The multigerm γ lives in the plane ℝ^{2}(x, y); it is tangent to Y. The quasi-homogeneity of γ in Theorem 4.1 follows from Propositions 4.2, 4.3 and 4.4 and the following statement proved in Section 4.5.

Proposition 4.5. Let γ be a multigerm of a curve in ℝ^{2}. In the C^{∞} category assume that γ contains no components with the zero Taylor series. If γ is tangent to a vector field Y on ℝ^{2} with non-zero real eigenvalues of the same sign at the singular point 0 ∈ ℝ^{2} then the multigerm γ is quasi-homogeneous.

4.2. Proof of the implication (iv) ⟹ (i). Let γ_{i} be any component of a quasi-homogeneous multigerms of a curve in ℝ^{3} with weights λ_{1}, λ_{2}, λ_{3} > 0 in local coordinates (x, y, z). Consider the 1-forms

θ_{1} = λ_{1}xdy − λ_{2}ydx, θ_{2} = λ_{1}xdz − λ_{3}zdx, θ_{3} = λ_{2}ydz − λ_{3}zdy.

The quasi-homegeneity of γ impleis that γ_{i}^{∗}θ_{1} = γ_{i}^{∗}θ_{2} = γ_{i}θ_{3} ≡ 0. Let α be the germ a contact 1-form with respect to which γ is Legendrain. Then γ_{i}^{∗}(α + rθ_{j}) ≡ 0, j = 1, 2, 3, for any r ∈ ℝ. Since the weights λ_{i} are positive and α(0) ≠ 0 then there exists at least one j ∈ {1, 2, 3} such that (α ∧ dθ_{j})(0) ≠ 0. We may assume that (α ∧ dθ_{1})(0) ≠ 0. Denote α_{r} = α + rθ_{1}. Since θ_{1}(0) = 0 then

(α_{r} ∧ dα_{r})(0) = (α ∧ dα)(0) + r(α ∧ dθ_{1})(0), r ∈ ℝ.

For sufficiently big r the 1-forms α_{r} and α_{−r} are contact and define different orientations of ℝ^{3}. The multigerm γ is Legendrain with respect to these contact 1-forms, and consequently γ does not determine orientation of ℝ^{3}.

4.3. Proof of Proposition 4.3. Take a local coordinate system in which the vector field Y_{α,β} vanishes at points of the z-axes and consequently has the form

f_{1}(x, y, z)\frac{∂}{∂x} + f_{2}(x, y, z)\frac{∂}{∂y} + f_{3}(x, y, z)\frac{∂}{∂z}

with f_{1}, f_{2} satisfying (4.3) and f_{3}(x, y, z) such that f_{3}(0, 0, z) ≡ 0. The eigenvalues at a singular point (0, 0, z) are λ_{1}(z), λ_{2}(z), 0, where λ_{1}(0) and λ_{2}(0) are real numbers of the same sign. The absence of resonant relations 0 = k_{1}λ_{1}(0) + k_{2}λ_{2}(0) with integers k_{1}, k_{2} ≥ 0, (k_{1}, k_{2}) ≠ (0, 0) implies that f_{3}(x, y, z) can be reduced to 0 by a formal change of coordinates, see [5]. The condition that λ_{1}(0) and λ_{2}(0) are real numbers of the same sign implies that normal form (3.2)-(3.3) also holds in the C^{∞} and real analytic categories, see [5] and [19].

4.4. Proof of Proposition 4.4. Let

γ_{i} : x = a(t), y = b(t), z = c(t)

be any component of γ. The tangency of Y_{α,β} to γ implies the relation

c′(t) · f_{1}(a(t), b(t), c(t)) = c′(t) · f_{2}(a(t), b(t), c(t)) ≡ 0.  (4.4)

Let â, b̂, ĉ be the Taylor series of a(t), b(t), c(t). The vector field Y_{α,β} has two nonzero eigenvalues and consequently

det (\frac{∂(f_{1}, f_{2})}{∂(x, y)}) (0) ≠ 0.  (4.5)

Relations (4.3), (4.4) and (4.5) imply that either ĉ = 0 or â = b̂ = 0. In the real analytic category we obtain the alternative (AL) c(t) ≡ 0 or a(t) = b(t) ≡ 0. The same alternative holds in the C^{∞} category due to the assumption γ ∉ E_{∞}. In fact, if one of the series â, b̂ is not zero then by (4.3), (4.4), (4.5) one has c(t) ≡ 0. On the other hand, if â = b̂ = 0 then the assumption γ ∉ E_{∞} implies that c′(0) ≠ 0 and (4.3),(4.4), (4.5) imply that a(t) = b(t) ≡ 0.

Now we will show that the case a(t) = b(t) ≡ 0 is impossible for any component of the multigerm γ satisfying (A). Take one of contact forms with respect to which γ is Legendrain, for example the 1-form α, and write it in local coordinates of normal form (4.2) − (4.3): α = α_{1}dx + α_{2}dy + α_{3}dz. Assume that γ contains a component x = 0, y = 0, z = c(t) ≢ 0. Then in the real analytic category α_{3}(0) = 0. The same holds in the C^{∞} category because the assumption γ ∉ E_{∞} implies c′(0) ≠ 0. Consequently (α_{1}(0), α_{2}(0)) ≠ (0, 0). Consider the vector field α_{1}(x, y, 0)\frac{∂}{∂y} − α_{2}(x, y, 0)\frac{∂}{∂x}. It is a non-singular vector field in the (x, y)-plane. Denote by l its phase curve containing the origin. Then l is a non-singular 1dimensional submanifold of ℝ^{3}. By (AL) any component of the multigerm γ belongs either to the z-axes or to the (x, y)-plane and it follows that the image of γ belongs to the union of l and the z-axes. This contradicts to assumption (A).

4.5. Proof of Proposition 4.5. We will use local normal forms for vector fields on the plane. It is well-known that the germ of a vector field Y satisfying the assumptions of Proposition 4.5 can be brought to one of the following normal forms

ẋ = λ_{1}x, ẏ = λ_{2}y;  (4.6)
ẋ = λx, ẏ = Nλy + κx^{N}, N ≥ 1, λ ≠ 0, κ ≠ 0.  (4.7)

This norml form holds in either the rela analytic or the C^{∞} category, see [5]. Proposition 4.5 is a direct corollary of the following lemma.

Lemma 4.6. Let γ : x = a(t), y = b(t) be the germ of a plane curve. In the C^{∞} category assume that at least one of the germs a(t), b(t) has non-zero Taylor series.

(i) If γ is tangent to a vector field of form (4.6) then, up to a reparameterization t → ϕ(t), ϕ′(0) ≠ 0, it the form

a(t) = a · t^{λ_{1}r,}, b(t) = b · t^{λ_{2}r}, r, a, b ∈ ℝ.  (4.8)

(ii) If γ is tangent to a vector field of form (4.7) then a(t) ≡ 0.

The proof of this lemma in the C^{∞} category requires the following known statement (its proof and the proof of much more general statements of the same type can be found, for example, in [Be]):

Lemma 4.7. Let Q(t, u) be a C^{∞} function-germ such that Q(t, 0) ≡ 0. Let f(t) be a C^{∞} function germ satisfying the equation t^{r} · f′(t) = Q(t, f(t)) for some r ≥ 0. If f(t) has the zero Taylor series then f(t) ≡ 0.

Proof of Lemma 4.6, (i). We may assume that a(t) has non-zero Taylor series. Then there is no loss of generality to assume that a(t) = at^{p}, a ≠ 0. The tangency of the curve (at^{p}, b(t)) to a vector field of form (4.6) means the relation

λ_{2}pb(t) = λ_{1}tb′(t).  (4.9)

This equation can be easily solved in formal power series. If λ_{2}p/λ_{1} is not an integer number then the only formal solution of (4.9) is b(t) ≡ 0. If λ_{2}p/λ_{1} is an integer number i then any formal solution of equation (4.9) has the form b(t) = bt^{i}, b ∈ ℝ. Therefore any analytic solution of (4.9) and, by Lemma 4.7, any C^{∞} solution of (4.9) has the form b(t) = bt^{λ_{2}p/λ_{1}}, where b = 0 if λ_{2}p/λ_{1} is not an integer number and b ∈ ℝ if λ_{2}p/λ_{1} is an integer. In any case we obtain (4.8) with r = p/λ_{1}. Proof of Lemma 4.6, (ii). At first let us show that the function germ a(t) has the zero Taylor series. Assume, to get contradiction, that this is not so. Then there is no loss of generality to assume that a(t) = at^{p}, a ≠ 0. The tangency of the curve (at^{p}, b(t)) to a vector field of form (4.7) means the relation

F (t) = λtb′(t) − pNλb(t) − pκa^{N}t^{pN} ≡ 0.

Let ∑ f_{i}t^{i} be the Taylor series of F (t). Then F_{Np} does not depend on b(t) and is equal to F_{Np} = −pκa^{n} ≠ 0. Contradiction.

We have proved that the function germ a(t) has the zero Taylor series. Consequently in the real analytic category a(t) ≡ 0. In order to prove Lemma 4.6 in the C^{∞} category we have to prove that a(t) ≡ 0 under the assumption that a(t) has the zero Taylor series and the Taylor series of b(t) is not zero. In this case there is no loss of generality to assume that b(t) = bt^{p}, b ≠ 0. The tangency of the curve (a(t), bt^{p}) to a vector field of form (4.7) means the relation

a′(t) · (Nλbt^{p} + κa^{N}(t)) = λpbt^{p−1}a(t).  (4.10)

Since the function germ a(t) has the zero Taylor series then a(t) = t^{p}ã(t) for some C^{∞} function germ ã(t). Equation (4.10) for a(t) is equivalent to a certain equation for ã(t) of the form t^{p}ã′(t) = Q(t, ã(t)), where Q is a C^{∞} function germ of two variables. By Lemma 4.7 ã(t) ≡ 0. Consequently a(t) ≡ 0.

5. Proof of Proposition 4.2

5.1. Outline of the proof. The starting point is the following lemma.

Lemma 5.1. Let (α) and (β) be tangent at the origin contact structures defining different orientations of ℝ^{3}. Then the vector field Y_{α,β} has the following properties:

(a) the sum of its eigenvalues at the origin is not zero;

(b) the ideal generated by the three coefficients of Y_{α,β} in some coordinate system (this ideal does not depend on the choice of the coordinate system) is generated by two function germs (and consequently one of the eigenvalues of Y_{α,β} is zero).

Lemma 5.1 is proved in Section 5.2. Note that the multigerm γ is not involved in Lemma 5.1. In Sect. 5.3 we will show that Proposition 4.2 follows from Lemma 5.1 and the following property of any Legendrain multigerm satisfying (A).

Proposition 5.2. Let γ = (γ_{1}, ..., γ_{m}) ∈ Leg(ℝ^{3}) be a multigerm satisfying assumption (A). In the C^{∞} category we assume that γ ∉ E_{∞}. There exists a local coordinate system (x, y, z) and a positive number r such that the following holds. If θ = θ_{1}dx + θ_{2}dy + θ_{3}dz is a 1-form such that γ_{i}^{∗}θ ≡ 0, i = 1, .., m then

θ_{1}(0) = θ_{2}(0) = 0, \frac{∂θ_{1}}{∂x} (0) = 0, \frac{∂θ_{1}}{∂y} (0) + r \frac{∂θ_{2}}{∂x} (0) = 0.  (5.1)

Proposition 5.2 is proved in Sections 5.4 - 5.6 for the case that the multigerm γ satisfies, except (A), at least one of the following assumptions: (B1) the multigerm γ contains three non-singular components such that no two of them are tangent at 0. (B2) the multigerm γ contains two tangent at 0 non-singular components; the order of tangency is finite. (B3) the multigerm γ contains a singular component whose Taylor series is not RL-equivalent to a series of the form (at^{r}, 0, 0), r ≥ 2.

One can easily check that in the C^{∞} category the validity of at least one of the assumptions (B1) - (B3) follows from (A) and the condition γ ∉ E_{∞}. In the real analytic category there are multigerms γ satisfying (A) and violating each of the assumptions (B1),(B2),(B3), but in Section 5.7 we reduce Theorem 5.2 to the case that at least one of these assumptions holds.

5.2. Proof of Lemma 5.1. We may assume that α(0) = β(0). Take local coordinates in which α = A_{1}dx + A_{2}dy + dz, β = B_{1}dx + B_{2}dy + dz, where A_{i}(0) = B_{i}(0) = 0. Calculate the vector field Y_{α,β} using the volume form Ω = dx ∧ dy ∧ dz:

Y_{α,β} = (A_{2} − B_{2})\frac{∂}{∂x} + (B_{1} − A_{1})\frac{∂}{∂y} + (A_{1}B_{2} − A_{2}B_{1})\frac{∂}{∂z}

Statement (i) follows from the relation A_{2}B_{1}−A_{1}B_{2} = (A_{2}−B_{2})·A_{1}+(B_{1}−A_{1})·A_{2}. The sum of the eigenvalues of Y_{α,β} is equal to

trace Y_{α,β} = \frac{∂(A_{2} − B_{2})}{∂x}(0)+ \frac{∂(B_{1} − A_{1})}{∂y}(0) = (\frac{∂A_{2}}{∂x} − \frac{∂A_{1}}{∂y})(0)−(\frac{∂B_{2}}{∂x} − \frac{∂B_{1}}{∂y})(0).

The assumption that the contact structures (α) and (β) define different orientations means that one of the numbers (\frac{∂A_{2}}{∂x} − \frac{∂A_{1}}{∂y})(0), (\frac{∂B_{2}}{∂x} − \frac{∂B_{1}}{∂y})(0) is positive and the other is negative. Consequently trace Y_{α,β} ≠ 0.

5.3. Lemma 5.1 and Proposition 5.2 imply Proposition 4.2. Let γ, α, β be as in Proposition 4.2. Take local coordinates as in Proposition 5.2. Let

α = A_{1}dx + A_{2}dy + A_{3}dz, β = B_{1}dx + B_{2}dy + B_{3}dz.

Then

A_{1}(0) = A_{2}(0) = B_{1}(0) = B_{2}(0) = 0, \frac{∂A_{1}}{∂x} (0) = \frac{∂B_{1}}{∂x} (0) = 0,  (5.2)
\frac{∂A_{1}}{∂y} (0) + r \frac{∂A_{2}}{∂x} (0) = 0, \frac{∂B_{1}}{∂y} (0) + r \frac{∂B_{2}}{∂x} (0) = 0.  (5.3)

It follows that A_{3}(0) ≠ 0, B_{3}(0) ≠ 0 and therefore there is no loss of generality to assume that A_{3} = B_{3} ≡ 1. The vector field Y_{α,β} has form (5.1), up to multiplication by a non-vanishing function. Relations (5.2) imply that the matrix of linear approximation of Y_{α,β} is a triangular matrix with diagonal entries (eigenvalues)

λ_{1} = \frac{∂(A_{2} − B_{2})}{∂x}(0), λ_{2} = \frac{∂(B_{1} − A_{1})}{∂y}(0), λ_{3} = 0.

Relations (5.3) imply that λ_{2} = rλ_{1}. The case λ_{1} = λ_{2} = 0 is impossible by the statement (a) of Lemma 5.1. Therefore the vector field Y_{α,β} has two non-zero real eigenvalues of the same sign. It remains to prove that the set of singular points of Y_{α,β} is a non-singular 1-dimensional submanifold of ℝ^{3}. This easily follows from the statement (b) in Lemma 5.1 and the fact that Y_{α,β} has two non-zero eigenvalues.

5.4. Proof of Proposition 5.2 under assumption (B1). Let γ_{1}, γ_{2}, γ_{3} be the components satisfying (B1). The condition γ ∈ Leg(ℝ^{3}) implies that the vectors γ′_{1}(0), γ′_{2}(0), γ′_{3}(0) are linearly dependent. In fact, if these vectors are linearly independent and α is a 1-form such that γ_{1}^{∗}α = γ_{2}^{∗}α = γ_{3}^{∗}α ≡ 0 then α(0) = 0, i.e. α is not contact. Thus the vectors γ′_{1}(0), γ′_{2}(0), γ′_{3}(0) are linearly dependent, but any two of them are linearly independent. Therefore in suitable local coordinates

γ′_{1}(0) = \frac{∂}{∂x}, γ′_{2}(0) = \frac{∂}{∂y} , γ′_{3}(0) = \frac{∂}{∂x} + \frac{∂}{∂y} .

Let θ = θ_{1}dx + θ_{2}dy + θ_{3}dz be a 1-form such that γ_{i}^{∗}θ ≡ 0, i = 1, 2, 3. Then one has the relations

θ_{1}(0) = θ_{2}(0) = 0, \frac{∂θ_{1}}{∂x} (0) = \frac{∂θ_{2}}{∂y} (0) = 0, \frac{∂θ_{1}}{∂y} (0) + \frac{∂θ_{2}}{∂x} (0) = 0.

Therefore Proposition 5.2 holds with the chosen local coordinates and with r = 1.

5.5. Proof of Proposition 5.2 under assumption (B2). Let γ_{1}, γ_{2} be the components satisfying (B2). In suitable coordinates their images can be described by equations ^{6}

y = z = 0, y − x^{N+1} = z = 0, N ≥ 1.  (5.4)

Let θ = θ_{1}dx + θ_{2}dy + θ_{3}dz be a 1-form such that γ_{1}^{∗}θ = γ_{2}^{∗}θ ≡ 0, where γ_{1}, γ_{2} have form (5.4). Then one has the relations

θ_{1}(x, 0, 0) ≡ 0, θ_{1}(x, x^{N+1}, 0) + (N + 1)x^{N}θ_{2}(x, x^{N+1}, 0) ≡ 0.

6Normal form (5.4) can be easily obtained as follows. The images of γ_{1} and γ_{2} can be described by equations y = z = 0 and y = f_{1}(x), z = f_{2}(x), where f_{i}(x) = x^{N+1}g_{i}(x), (g_{1}(0), g_{2}(0)) ≠ (0, 0).

Here N is the order of tangency between γ_{1} and γ_{2}. Applying a change of coordinates of the form (x, y, z) → (ϕ(x), ±y, ±z) and possibly (x, y, z) → (x, z, y) we can reduce f_{1}(x) to x^{N+1} and f_{2}(x)

to x^{N+1}f̃_{2}(x). To get (5.4) it remains to change z by z − yf̃_{2}(x).

Considering the coefficients at x^{i}, i = 0, 1, N, N + 1 in the Taylor series of the left hand side of this relation we obtain

θ_{1}(0) = θ_{2}(0) = \frac{∂θ_{1}}{∂x} (0) = 0, \frac{∂θ_{1}}{∂y} (0) = (N + 1)\frac{∂θ_{2}}{∂x} (0).

Therefore Proposition 5.2 holds in the chosen coordinates r = N + 1.

5.6. Proof of Proposition 5.2 under assumption (B3). Let γ_{1} be a singular component satisfying (B3). Then γ_{1} is RL-equivalent to a curve of the from

ψ : x = t^{p}, y = t^{q} + f_{1}(t), z = f_{2}(t),  (5.5)
2 ≤ q < p, q ≠ 0 mod p, f_{1}(t) = o(t^{q}), f_{2}(t) = o(t^{q}).  (5.6)

We will assume that the functions f_{1}(t) and f_{2}(t) satisfy the following conditions

f_{1}^{(2p)}(0) = f_{2}^{(2p)}(0) = f_{2}^{(p+q)}(0) = 0.  (5.7)

These conditions can be reached by a change of coordinates of the form y → y + a_{1}x^{2}, z → z + a_{2}x^{2} + a_{3}xy with suitable a_{1}, a_{2}, a_{3}. Let θ = θ_{1}(x, y, z)dx + θ_{2}(x, y, z)dy+θ_{3}(x, y, z)dz be a 1-form such that ψ^{∗}θ ≡ 0. Then one has the relation

pt^{p−1}θ_{1}(ψ̂(t)) + (q · t^{q−1} + f′_{1}(t))θ_{2}(ψ̂(t)) + f′_{2}(t)θ(ψ̂(t)) ≡ 0.

Using (5.6) and (5.7) we see that the coefficients at t^{p−1}, t^{q−1}, t^{2p−1} and t^{p+q−1} in the Taylor series of the left hand side of this relation are equal to p · θ_{1}(0), q · θ_{2}(0), p · \frac{∂θ_{1}}{∂x} (0) and p · \frac{∂θ_{1}}{∂y} (0) + q · \frac{∂θ_{2}}{∂x} (0) respectively. Therefore

θ_{1}(0) = θ_{2}(0) = \frac{∂θ_{1}}{∂x} (0), p · \frac{∂θ_{1}}{∂y} (0) + q · \frac{∂θ_{2}}{∂x} (0) = 0,

and Proposition 5.2 holds with the chosen coordinate system and with r = q/p.

5.7. Real analytic category. Reduction of Proposition 5.2 to the case that one of the assumptions (B1) - (B3) holds. Let γ = (γ_{1}, ..., γ_{m}). Construct a multigerm γ̃ = (γ̃_{1}, ..., γ̃_{m}) as follows. If γ_{i} is a non-singular component then γ̃_{i} = γ_{i}. If γ_{i} is a singular component whose image is not contained in in a nonsingular 1-dimensional submanifold of ℝ^{3} then also γ̃_{i} = γ_{i}. In the remaining case that γ_{i} is a singular component whose image is contained in a non-singular 1-dimensional submanifold l_{i} of ℝ^{3} we define γ̃_{i} to be a non-singular curve whose image is l_{i}. Let θ be any 1-form. It is clear that in the real analytic category the relation γ_{i}^{∗}θ ≡ 0 implies the relation γ̃_{i}^{∗}θ = 0. Therefore in order to prove Proposition 5.2 for the multigerm γ it suffices to prove it for the multigerm γ̃.

Since the image of γ is contained in the image of γ̃ then the multigerm γ̃ also satisfies A. Let us show that γ̃ also satisfies at least one of the assumptions (B1) - (B3). The construction of γ̃ implies that it satisfies one of the following conditions:

(a) γ̃ contains non-singular components only;

(b) γ̃ contains a singular component whose image is not contained in a non-singular 1-dimensional submanifold of ℝ^{3}.

In case (a) the image of γ̃ is the union l_{1} ∪ ⋯ ∪ l_{s} of s different non-singular 1-dimensional submanifolds of ℝ^{3} (it is not excluded that s < m). By assumption A s ≥ 3. In the real analytic category the order of tangency between two different non-singular components of a multigerm is always finite. It follows that in case (a) either (B1) or (B2) holds. In case (b) one has (B1) by the following lemma.

Lemma 5.3. (real analytic category). Let ψ : (ℝ, 0) → (ℝ^{3}, 0) be the germ of a singular curve whose Taylor series ψ̂ is RL-equivalent to a series of the form (t^{r}, 0, 0), r ≥ 2. Then ψ is RL-equivalent to the curve (t^{r}, 0, 0).

Proof. There is no loss of generality to assume that ψ : x = t^{r}, y = b(t), z = c(t). Let b̂(t), ĉ(t) be the Taylor series of b(t) and c(t). The RL-equivalence of ψ̂ to (t^{r}, 0, 0) implies that there exist formal power series B̂(x, y, z), Ĉ(x, y, z) such that b̂(t) = B̂(t^{r}, 0, 0), ĉ(t) = Ĉ(t^{r}, 0, 0). It follows that b(t) = g(t^{r}), c(t) = h(t^{r}), where g and h are analytic functions in one variable. A local diffeomorphism (x, y, z) → (x, y + g(x), z + h(x)) brings the curve (t^{r}, 0, 0) to the curve ψ. □

6. Contact classification of A singularities

The simplest application of results of section 1 is the classification of singular Legendrian submanifolds of a contact 3-space diffeomorphic to submanifolds

A_{i} : {(x, y, z) : y^{2} − x^{i+1} = z = 0.}

Submanifolds of ℝ^{3} diffeomorphic to A_{i} correspond to the following singularity classes of multigerms of curves in ℝ^{3}: A_{2}∪A_{4} ∪⋯: singular curves ψ with non-zero 2-jet whose Taylor series is not RLequivalent to (t^{2}, 0, 0); ψ ∈ A_{2k} if j^{2k_{1}}ψ is RL-equivalent to (t^{2}, 0, 0) and j^{2k+1}ψ is not equivalent to (t^{2}, 0, 0)); A_{1}∪A_{3} ∪ ⋯: multigerms γ = (γ_{1}, γ_{2}) consisting of two non-singular components with finite order of tangency; γ ∈ A_{2k+1} if the order of tangency is equal to k (the zero order of tangency means that γ_{1} and γ_{2} are not tangent).

The classes A_{2k} and A_{2k+1} are described by the following normal form with respect to the RL-equivalence:

Â_{2k} : (t^{2}, t^{2k+1}, 0), Â_{2k+1} : (t, 0, 0), (t, t^{k+1}, 0)

see [1] an Sect. 5.5. The image of Â_{2k} is exactly A_{2k}. The image of Â_{2k+1} is given by the equations y(y − x^{k+1}) = z = 0. The function y(y − x^{k+1}) is R-equivalent to y^{2} − x^{2k+2}. Therefore the image of Â_{2k+1} is diffeomorphic to A_{2k+1}.

Fix a contact structure, for example the standard contact structure (dz − ydx). Consider the following singular curve and a multigerm with two components which are Legendrain with respect to this contact structure:

Ã_{2k} : x = t^{2}, y = t^{2k+1}, z = 2t^{2k+3}/(2k + 3);

Ã_{2k+1} : γ_{1} : x = t, y = z = 0, γ_{2} : x = t, y = t^{k+1}, z = t^{k+2}/(k + 2).

These multigerms can be reduced to Â_{2k} and Â_{2k+1} by a change (x, y, z) → (x, y, z+ rxy) with a suitable r. Therefore Theorem 1.1 implies the following result.

Theorem 6.1. (real analytic and C^{∞} categories). Let i = 2k or i = 2k + 1.

1. A_{i} ∩ Leg(ℝ^{3}) =A_{i}.

2. Let γ ∈A_{i} be a Legendrain multigerm in a fixed contact structure (ℝ^{3}, (dz−ydx)). Then γ is contactomorphic to the multigerm Ã_{i}.

By Theorem 6.1 all Legendrain multigerms of the class A_{i} in a fixed contact space are contactomorphic. Consequently, all singular Legendrian submanifolds diffeomorphic to A_{i} are contactomorphic. We obtain the following corollary.

Corollary 6.2. (real analytic and C^{∞} categories). Any singular Legendrian submanifold of the contact space (ℝ^{3}, (dz − ydx)) diffeomorphic to A_{i} is contactomorphic to the singular submanifold given by the equations y^{2} = x^{i+1}, z = 2xy/(i + 3).

7. Contact classification of E singularities

In this section we give a complete contact classification of germs of Legendrain singular curves in ℝ^{3} of class E. The class E consists of germs ψ : (ℝ, 0) → (ℝ^{3}, 0) satisfying the following conditions: (a) ψ has zero 2-jet and non-zero 3-jet; (b) the Taylor series of ψ is not RL-equivalent to (t^{3}, 0, 0).

In the previous section we showed that A∩Leg(ℝ^{3}) = A.F orE singularities this is not so: within such singularities there are those that are not Legendrian with respect to any contact structure. The class E∩Leg(ℝ^{3}) is substantially smaller than E. For example, the classification of E singularities with respect to diffeomorphisms starts with the following four singularities, see [9]:

(t^{3}, t^{4}, t^{5}) ← (t^{3}, t^{4}, 0)^{∗} ← (t^{3}, t^{5}, t^{7}) ← (t^{3}, t^{5}, 0)^{∗}.

Here we marked by ∗ the two singularities that belong to Leg(ℝ^{3}). The other two singularities do not belong to Leg(ℝ^{3}) by Theorem 1.3: they are quasi-homogeneous, but not planar. The ierarchy of E singularities continues with

(t^{3}, t^{5}, t^{7})

↑ (t^{3}, t^{7}, t^{8}) ← (t^{3}, t^{7} + t^{8}, t^{11})^{∗} ← (t^{3}, t^{7} + t^{8}, 0)

↑ ↑ (t^{3}, t^{8}, t^{10}) (t^{3}, t^{7}, t^{11}).

Within these six singularities only one belongs to Leg(ℝ^{3}) (t^{3}, t^{7} + t^{8}, t^{11}) ∈ Leg(ℝ^{3}) by Proposition 2.1. The other five singularites do not belong to Leg(ℝ^{3}) by Theorem 1.3 - each of them is either quasi-homogeneous and not planar, or planar and not quasi-homogeneous.

Now we present a general result on distinguishing the class E ∩ Leg(ℝ^{3}) in E. The classification of all E singularities with respect to the RL-equivalence is known due to [9], see also [2]. Any singular curve of class E is RL equivalent to one and only one of the following curves, up to reduction ± → + given in the remark below (we use the notations in [2]):

E_{6k,p,i}^{±} : x = t^{3}, y = t^{3k+1} ± t^{3k+2+3i}, z = t^{3k+2+3p},

E_{6k+2,p,i}^{±} : x = t^{3}, y = t^{3k+2} ± t^{3k+4+3i}, z = t^{3k+4+3p},

where k ≥ 1, 0 ≤ p ≤ k, 0 ≤ i ≤ min(k − 1, p).

Remark. The curves E_{6k,p,i}^{±} (respectively E_{6k+2,p,i}^{±}) are diffeomorphic if and only if one of the following holds: (a) i is an even number (respectively odd number);

(b) i = k − 1; (c) i = p.

Proposition 7.1. Let l = 6k or l = 6k+2. E_{l,p,i}^{±} ∈ Leg(ℝ^{3}) if and only if p = i+1.

Proof. It is easy to check that the E singularities have the following properties:

• if p = k, i = k − 1 then the curves E_{6k,p,i}^{±} and E_{6k+2,p,i}^{±} are planar and quasihomogeneous (these curves are RL-equivalent to (t^{3}, t^{3k+1}, 0) and (t^{3}, t^{3k+2}, 0)); • if p = k, i ≤ k − 2 then the curves E_{6k,p,i}^{±} and E_{6k+2,p,i}^{±} are planar, but not quasi-homogeneous (these curves are RL-equivalent to (t^{3}, t^{3k+1} ± t^{3k+2+3i}, 0) and (t^{3}, t^{3k+2} ± t^{3k+4+3i}, 0));

• if i = p ≤ k − 1 then the curves E_{6k,p,i}^{±} and E_{6k+2,p,i}^{±} are quasi-homogeneous, but not planar (these curves are RL-equivalent to (t^{3}, t^{3k+1}, t^{3k+2+3p}) and (t^{3}, t^{3k+2}, t^{3k+4+3p}));

• in the remaining case i < p ≤ k − 1 the curves E_{6k,p,i}^{±} and E_{6k+2,p,i}^{±} are neither planar, nor quasi-homogeneous.

In the cases (a) p = k and (b) i = p Proposition 7.1 is a corollary of Theorem 1.3 and the given above properties of the E singularities. In the remaining case i < p ≤ k − 1 Proposition 7.1 follows from Proposition 2.1. □

Consider now the following curves of the class E that are Legendrain with respect to the contact structure (dz − ydx):

Ẽ_{6k,i+1,i}^{±} : x = t^{3}, y = t^{3k+1} ± t^{3k+2+3i}, z = ∫_{0}^{t} y(s)x′(s);

Ẽ_{6k+2,i+1,i}^{±} : x = t^{3}, y = t^{3k+2} ± t^{3k+4+3i}, z = ∫_{0}^{t} y(s)x′(s), where k ≥ 1, 0 ≤ i ≤ k − 1.

Theorem 7.2. (real analytic and C^{∞} categories). Any germ ψ ∈E of a Legendrain curve in the contact space (ℝ^{3}, dz − ydx)) is contactomorphic to one of the curves Ẽ_{6k,i+1,i}^{±}, Ẽ_{6k+2,i+1,i}^{±}, where k ≥ 1, 0 ≤ i ≤ k − 1. Within these curves no two are contactomorphic up to reduction ± → + in the given above remark. The curves Ẽ_{6k,k,k−1}^{±}, Ẽ_{6k+2,k,k−1}^{±} are planar and quasi-homogeneous. The curves Ẽ_{6k,i+1,i}^{±}, Ẽ_{6k+2,i+1,i}^{±} with i ≤ k − 2 are neither planar nor quasi-homogeneous.

Theorem 7.2 follows from Proposition 7.1 and Theorem 1.1: it suffices to observe that the curves Ẽ_{6k,i+1,i}^{±} and Ẽ_{6k+2,i+1,i}^{±} can be reduced to E_{6k,i+1,i}^{±} and E_{6k+2,i+1,i}^{±} by a change of the form z → k_{1}z + k_{2}xy with suitable k_{1}, k_{2}.

8. Contact-simple germs of Legendrain curves

Denote by MG^{m}(ℝ^{n}) the space of mulrigerms (γ_{1}, ..., γ_{m}), where γ_{i} : (ℝ, 0) → (ℝ^{n}, 0) are germs of curves in ℝ^{n}.

Definition. A multigerm γ ∈ MG^{m}(ℝ^{n}) is called simple in MG^{m}(ℝ^{n}) if there exist a set N ⊂ MG^{m}(ℝ^{n}) consisting of a finite number of multigerms and a finite k such that any multigerm in MG^{m}(ℝ^{n}) whose k-jet is sufficiently close to the k-jet of γ is diffeomorphic to one of the multigerms of the set N.

The simple multigerms in MG^{m}(ℝ^{n}) were determined and classified in [8] for m = 1, n = 2, in [9] for m = 1, n = 3, and recemtly in [13] for any m and n. Notation. Given a contact structure (α) on ℝ^{3} denote by Leg^{m}(ℝ^{3}, (α)) the subclass of MG^{m}(ℝ^{3}) consisting of multigerms of Legendrain curves in the contact space (ℝ^{3}, (α)). Definition. Let (α) be a contact structure on ℝ^{3}. A multigerm γ ∈ Leg^{m}(ℝ^{3}, (α)) is called contact-simple in Leg^{m}(ℝ^{3}, (α)) if there exists a set N ⊂ Leg^{m}(ℝ^{3}, (α)) consisting of a finite number of multigerms and a finite k such that any multigerm in Leg^{m}(ℝ^{3}, (α)) whose k-jet is sufficiently close to the k-jet of γ is contactomorphic to one of the multigerms of the set N.

In this section we use results of Section 1 in order to determine and classify all RL-contact-simple germs in Leg^{1}(ℝ^{3}, (α)). Most of results of the present section were obtained in [17] by a different method.

Theorem 8.1. Let (α) be any contact structure on ℝ^{3}. A germ ψ ∈ Leg^{1}(ℝ^{3}, (α)) is contact-simple in Leg^{1}(ℝ^{3}, (α)) if and only if it is simple in MG^{1}(ℝ^{3}).

Conjecture. The same holds when replacing Leg^{1}(ℝ^{3}, (α)) and MG^{1}(ℝ^{3}) by Leg^{m}(ℝ^{3}, (α) and MG^{m}(ℝ^{3}), for any m ≥ 1.

The part ”if” of Theorem 8.1 and of the conjecture follow from Theorem 1.1. The part ”only if” of Theorem 8.1 can be easily reduced to the following statement.

Proposition 8.2. Let (α) be any contact structure on ℝ^{3}. Let ψ_{1}, ψ_{2} ∈ Leg^{1}(ℝ^{3}, (α)) be germs of the form

ψ_{1} = (t^{4}, t^{9}, 0) + o(t^{9}), ψ_{2} = (t^{5}, t^{6}, 0) + o(t^{6}).

The germs ψ_{1} and ψ_{2} are not contact-simple in Leg^{1}(ℝ^{3}, (α)).

Proof. It is easy to check that the condition ψ_{1}, ψ_{2} ∈ Int(ℝ^{3}) implies that these germs have the following form:

ψ_{1} = (t^{4}, t^{9} + a_{1}t^{10} + a_{2}t^{11}, 0) + o(t^{11}; ψ_{2} = (t^{5}, t^{6} + b_{1}t^{7} + b_{2}t^{8} + b_{3}t^{9}, 0) + o(t^{9}).

The assumption ψ_{1}, ψ_{2} ∈ Leg^{1}(ℝ^{3}, (α)) implies that the contact 1-form α = Adx + Bdy + Cdz satisfies the condition C(0) ≠ 0 (here x, y, z are the first, the second, and the third coordinates.) Using this observation it is easy to show that there exist families ψ_{1,ϵ}, ψ_{2,ϵ} ∈ Leg^{1}(ℝ^{3}, (α)) of the form

ψ_{1,ϵ} = (t^{4}, t^{9} + a_{1}t^{10} + (a_{2} + ϵ)t^{11}, t^{12} · f_{1}(ϵ, t));
ψ_{2,ϵ} = (t^{5}, t^{6} + b_{1}t^{7} + b_{2}t^{8} + (b_{3} + ϵ)t^{9}, t^{10} · f_{2}(ϵ, t)),

where f_{1}(ϵ, t) and f_{2}(ϵ, t) are smooth functions. Now Proposition 8.2 is a corollary of the following classification results (see [8] and [9]): if ϵ and ϵ̃ are small enough then the 11-jets j^{11}ψ_{1,ϵ} and j^{11}ψ_{1,ϵ̃} (respectively the 9-jets j^{9}ψ_{2,ϵ} and j^{9}ψ_{2,ϵ}) are RL-equivalent if and only if ϵ = ϵ̃. □

Theorem 8.1 and results in [9] imply the following corollary.

Theorem 8.3. A germ ψ ∈ Leg^{1}(ℝ^{3}, (α)) is contact-simple in Leg^{1}(ℝ^{3}, (α)) if and only if it it satisfies each of the following requirements: (a) the 4-jet of ψ is not zero; (b) the 9-jet of ψ is not RL-equivalent to (t^{4}, t^{9}) or (t^{4}, 0); (c) the Taylor series of ψ is not RL-equivalent to one of the series (t^{2}, 0, 0), (t^{3}, 0, 0), (t^{4}, t^{6}, 0).

In order to obtain the classification of the contact-simple curves in Leg^{1}(ℝ^{3}, (α) one has to take out from the list of simple singularities in MG^{1}(ℝ^{3}) all curves which are not Legendrain with respect to any contact structure.

Theorem 8.4. Let (α) be any contact structure on ℝ^{3} and let ψ ∈ Leg^{1}(ℝ^{3}, (α)) be a non-immersed contact-simple germ in Leg^{1}(ℝ^{3}, (α)). Then one has the following alternative:

(a) the germ ψ is planar, quasi-homogeneous and RL-equivalent to one and only one of the germs (t^{2}, t^{2k+1}, 0), (t^{3}, t^{3k+1}, 0), (t^{3}, t^{3k+2}, 0), (t^{4}, t^{5}, 0), (t^{4}, t^{7}, 0), k ≥ 1;

(b) the germ ψ is neither planar nor quasihomogeneous and RL-equivalent to one and only one of the germs (t^{3}, t^{3k+1} ± t^{3k+2+3i}, t^{3k+5+3i}), k ≥ 2, i ∈ {0, ..., k − 2}, ± ↪ + if i is odd; (t^{3}, t^{3k+2} ± t^{3k+4+3i}, t^{3k+7+3i}), k ≥ 2, i ∈ {0, ..., k − 2}, ± ↪ + if i is even; (t^{4}, t^{6} + t^{2k+5}, t^{2k+9}), k ≥ 1; (t^{4}, t^{5} ± t^{7}, t^{11}), (t^{4}, t^{7} ± t^{9}, t^{13}), (t^{4}, t^{7} ± t^{13}, t^{17}).

Proof. For curve germs with non-zero 3-jet this theorem was already proved in Sections 6, 7. Consider curve germs with the 4-jet (t^{4}, 0, 0) and satisfying requirements b) - d) of Theorem 8.3. By classification results in [9] any such curve germ ψ satisfies one of the following conditions: a) ψ is planar, but not quasi-homogeneous;

b) ψ is quasi-homogeneous, but not planar; c) ψ is planar, quasi-homogeneous and RL-equivalent to one and only one of the curves (t^{4}, t^{5}, 0), (t^{4}, t^{7}, 0); d). ψ is neither planar, nor quasi-homogeneous and RL-equivalent to one and only one of the following curves:

(t^{4}, t^{6} + t^{2k+5}, t^{2k+7}), (t^{4}, t^{6} + t^{2k+5}, t^{2k+9}), (t^{4}, t^{6} + t^{2k+5}, t^{2k+13}), k ≥ 1;

(t^{4}, t^{5} ± t^{7}, t^{6}), (t^{4}, t^{5} ± t^{7}, t^{11}), (t^{4}, t^{7}, t^{9} + t^{10});
(t^{4}, t^{7} ± t^{9}, t^{10}), (t^{4}, t^{7} ± t^{9}, t^{13}), (t^{4}, t^{7} ± t^{9}, t^{17}), (t^{4}, t^{7} ± t^{13}, t^{17}).

In case c) ψ ∈ Leg(ℝ^{3}). By Theorem 1.3 in cases a) and b) ψ ∉ Leg(ℝ^{3}). In order to distinguish germs in Leg(ℝ^{3}) within case d) we use Proposition 2.1. By this proposition the curves (t^{4}, t^{6}+t^{2k+5}, t^{2k+9}), (t^{4}, t^{5}±t^{7}, t^{11}), (t^{4}, t^{7}±t^{9}, t^{13}), (t^{4}, t^{7}± t^{13}, t^{17}) belong to Leg(ℝ^{3}) and the other displayed curves do not belong to Leg(ℝ^{3}). Now Theorem 8.4 follows from Theorem 1.1. □

Now we establish a 1-1 correspondence between the contact classification of contact-simple curves and the RL-classification of simple germs in MG^{1}(ℝ^{2}) (i.e. germs of plane curves.) The latter classification was obtained in [8]. Any simple germ of a plane curve is RL equivalent to one and only one of the following curves:

(t^{2}, t^{2k+1}), (t^{3}, t^{3k+1}), (t^{3}, t^{3k+2}), (t^{4}, t^{5}), (t^{4}, t^{7});  (8.1)
(t^{3}, t^{3k+1} ± t^{3k+2+3i}), k ≥ 2, i ∈ {0, ..., k − 2}, ± ↪ + if iis odd;  (8.2)
(t^{3}, t^{3k+2} ± t^{3k+4+3i}), k ≥ 2, i ∈ {0, ..., k − 2}, ± ↪ + if i is even;  (8.3)
(t^{4}, t^{6} + t^{2k+5}), (t^{4}, t^{5} ± t^{7}), (t^{4}, t^{7} ± t^{9}), (t^{4}, t^{6} ± t^{13}), k ≥ 1  (8.4)

The correspondence between the normal forms in Theorem 8.4 and normal forms (10.1)-(10.4) is not canonical. It requires to fix a local coordinate system in which the contact structure (α) has the Darboux normal form (dz − ydx). Notations. Given a germ ψ = (x(t), y(t), z(t)) ∈ Leg^{1}(ℝ^{3}, (dz − ydx)) denote by π_{x,y}(ψ) the plane curve germ (x(t), y(t)). Given a plane curve germ μ : (x(t), y(t)) denote by π_{x,y}^{−1}(μ) the (unique) germ ψ ∈ Leg^{1}(ℝ^{3}, (dz − ydx) such that π_{x,y}ψ = μ.

Theorem 8.5.

1. A germ ψ ∈ Leg^{1}(ℝ^{3}, (dz − ydx)) is contact-simple in Leg^{1}(ℝ^{3}, (dz − ydx)) if and only if the plane curve germ π_{x,y}(ψ) is simple in MG^{1}(ℝ^{3}).

2. Let ψ, ψ̃ ∈ Leg^{1}(ℝ^{3}, (dz−ydx)) be contact-simple germs in Leg^{1}(ℝ^{3}, (dz−ydx)). The germs ψ, ψ̃ are contactomorphic if and only if the plane curve germs π_{x,y}(ψ), π_{x,y}(ψ̃) are RL-equivalent;

3. If ψ ∈ Leg^{1}(ℝ^{3}, (dz − ydx)) is a contact-simple non-immersed germ in Leg^{1}(ℝ^{3}, (dz − ydx)) then one has the following alternative:

(a) ψ is planar, quasi-homogeneous and contactomorphic to one and only one germ of the form π_{x,y}^{−1}(μ), where μ is a germ in the list (8.1).

(b) ψ is neither planar, nor quasi-homogeneous and contactomorphic to one and only one germ of the form π_{x,y}^{−1}(μ), where μ is a germ in the list (8.2)-(8.4).

Proof. Theorem 8.5 is a corollary of Theorem 8.4, Theorem 1.1, classification results for plane curves in [8] and the following two statements following from the classification results for space curves in [9]: a). any curve given in Theorem 8.4 is RL-equivalent to a curve π_{x,y}^{−1}(μ), where μ is one of the curves in the list (8.1) - (8.4); b). if μ is a curve in the list (8.1) - (8.4) then the curve π_{x,y}^{−1}(μ) is RL-equivalent to one of the curves given in Theorem 8.4 □

9. Contact classification of multigerms consisting of a cusp and a non-singular curve.

In this section we use results of section 1 in order to classify, in a fixed contact 3-space, multigerms γ = (γ_{1}, γ_{2}), where γ_{2} is a non-singular curve and γ_{1} is a singular curve with the cusp singularity - a curve diffeomorphic to A_{2k} : (t^{2}, t^{2k+1}, 0).

Notation. The class of such multigerms will be denoted (A_{2k},l).

Let us start with the case k = 1. The RL-classification of multigerms in (A_{2},l) is described by the following normal forms obtained in [13]:

a). γ_{1} : (t^{2}, t^{3}, 0), γ_{2} : (0, 0, t) b). γ_{1} : (t^{2}, t^{3}, 0), γ_{2} : (0, t, 0) c). γ_{1} : (t^{2}, t^{3}, 0), γ_{2} : (t, 0, t^{2}) d). γ_{1} : (t^{2}, t^{3}, 0), γ_{2} : (t, 0, 0)

In order to distinguish these normal forms denote by L(γ_{1}) ⊂ T_{0}ℝ^{3} the tangent plane to a non-singular surface containing the image of γ_{1} (all such non-singular surfaces have the same tangent plane at 0) and by l(γ_{1}) ⊂ L(γ_{1}) the 1-dimensional subspace of L(γ_{1}) which is the limit tangent line to γ_{1}. If γ_{1} is given by the normal form x = t^{2}, y = t^{3}, z = 0 then L(γ_{1}) is spanned by the vectors \frac{∂}{∂x} and \frac{∂}{∂y} and l(γ_{1}) is spanned by the vector \frac{∂}{∂x}. Normal form a) holds if γ_{2} is transversal to the plane L(γ_{1}). If γ_{2} is tangent to L(γ_{1}), but not tangent to l(γ_{1}) then one has normal form b). Normal forms c) and d) hold if γ_{2} is tangent to l(γ_{1}).

Normal forms a) - d) imply that any multigerm γ ∈(A_{2},l) is quasi-homogeneous (the weights are (2, 3, 1) in the case of normal forms a), b), d) and (2, 3, 4) in the case of normal form (d)). It is easy to check that multigerms a) and c) are not planar. Therefore b), d) ∈ Leg(ℝ^{3}) and by Theorem 1.3 a), c) ∉ Leg(ℝ^{3}). . We obtain the following statement.

Proposition 9.1.

1. Any multigerm of the class (A_{2},l) ∩ Leg(ℝ^{3}) is planar and quasi-homogeneous.

2. Let γ = (γ_{1}, γ_{2}) ∈ (A_{2},l) ∩ Leg(ℝ^{3}). If γ_{2} is not tangent to the limit tangent line l(γ_{1}) then γ is RL-equivalent to multigerm b). If γ_{2} is tangent to the limit tangent line l(γ_{1}) then γ is RL-equivalent to multigerm d).

Introduce the multigerms

γ_{NT} : γ_{1} : x = t^{2}, y = t^{3}, z = 2t^{5}/5, γ_{2} : x = 0, y = t, z = 0; γ_{T} : γ_{1} : x = t^{2}, y = t^{3}, z = 2t^{5}/5, γ_{2} : x = t, y = 0, z = 0.

These multigerms are Legendrain with respect to the contact structure (dz − ydx). They can be brought to multigerms b) and d) respectively by a change of coordinates (x, y, z) → (x, y, z + rxy) with a suitable r ∈ ℝ. Therefore Theorem 1.1 along with Proposition 9.1 imply the following classification result.

Theorem 9.2. (real analytic and C^{∞} categories). Let γ = (γ_{1}, γ_{2}) ∈(A_{2},l) be an Legendrain multigerm in a fixed contact space (ℝ^{3}, (dz − ydx)). Then γ is contactmorphic to one of the multigerms γ_{NT} , γ_{T}. The normal form γ_{NT} (respectively γ_{T}) holds if γ_{2} is not tangent (respectively tangent) to the limit tangent line l(γ_{1}).

The contact classification of multigerms of classes (A_{2k},l), k ≥ 2 is more involved: if k ≥ 2 then the set (A_{2k},l) ∩ Leg(ℝ^{3}) contains non-planar multigerms.

Any multigerm γ = (γ_{1}, γ_{2}) ∈(A_{2k},l) is RL-equivalent to a multigerm of one of the following forms, see [13]:

A. γ_{1} : (t^{2}, t^{2k+1}, 0), γ_{2} : (0, 0, t); B. γ_{1} : (t^{2}, t^{2k+1}, 0), γ_{2} : (0, t, 0); C. γ_{1} : (t^{2}, t^{2k+1}, 0), γ_{2} : (t, t^{i}, t^{d}), 2 ≤ i ≤ 2k − 1, d ≥ i + 1; D. γ_{1} : (t^{2}, t^{2k+1}, 0), γ_{2} : (t, 0, t^{d}), d ≥ 2.

Normal form A holds if γ_{2} is transversal to L(γ_{1}), the normal form B holds if γ_{2} is tangent to L(γ_{1}), but not tangent to l(γ_{1}). If γ_{2} is tangent to l(γ_{1}) then one has one of the normal forms C, D. Here L(γ_{1}) and l(γ_{1}) are the plane and the line in T_{0}ℝ^{3} defined in the same way as in the case k = 1.

Multigerms A, B and D are quasi-homogeneous (the weights are (2, 2k + 1, 1) in cases A and B and (2, 2k + 1, d(2k + 1)) in case D). Multigerm A is not planar. Multigerm B is planar. One can check that multigerm D is planar if and only if d ≥ 2k + 1. In this case and in this case only D is RL-equivalent to the multigerm γ_{1} : (t^{2}, t^{2k+1}, 0), γ_{2} : (s, 0, 0). By Corollary 1.3 one has A ∉ Leg(ℝ^{3}), B ∈ Leg(ℝ^{3}) and D ∈ Leg(ℝ^{3}) if and only if d ≥ 2k + 1.

Lemma 9.3. C ∈ Leg(ℝ^{3}) if and only if d = i + 1.

This lemma is proved below. Now we have a complete description of the class (A_{2k},l) ∩ Leg(ℝ^{3}).

Proposition 9.4. A multigerm γ ∈(A_{2k},l) belongs to Leg(ℝ^{3}) if and only if γ is RL-equivalent to one of the multigerms

γ_{1} : (t^{2}, t^{2k+1}, 0), γ_{2} : (0, t, 0);  (9.1)
γ_{1} : (t^{2}, t^{2k+1}, 0), γ_{2} : (t, t^{i}, t^{i+1}), 2 ≤ i ≤ 2k − 1;  (9.2)
γ_{1} : (t^{2}, t^{2k+1}, 0), γ_{2} : (t, 0, 0).  (9.3)

These normal forms can be joined to the following one-index family:

(A_{2k}, l))^{i} : γ_{1} : (t^{2}, t^{2k+1}, 0), γ_{2} : (t, t^{i}, t^{i+1}), 1 ≤ i ≤ 2k

(the boundary values of the index i correspond to (9.1) and (9.3): (A_{2k}, l))^{1} is RL-equivalent to (9.1) and (A_{2k}, l))^{2k} is RL-equivalent to (9.3)). Introduce the following multigerms:

(A_{2k}, l))_{∗}^{i}, 1 ≤ i ≤ 2k: γ_{1} : x = t^{2}, y = t^{2k+1}, z = 2t^{2k+3}/(2k + 3),

γ_{2} : x = t, y = t^{i}, z = t^{i+1}/(i + 1).

These multigerms are Legendrain with respect to the contact structure (dz−ydx).

Theorem 9.5. (real analytic and C^{∞} categories). Let γ = (γ_{1}, γ_{2}) ∈(A_{2k},l) be a Legendrain multigerm in a fixed contact space (ℝ^{3}, (dz − ydx)). Then γ is contactmorphic to one and only one of the multigerms (A_{2k}, l))_{∗}^{i}, 1 ≤ i ≤ 2k.

Theorem 9.5 is a corollary of Proposition 9.4 and Theorem 1.1 because the multigerms (A_{2k}, l))_{∗}^{i} can be reduced to (A_{2k}, l))^{i} by a change of coordinates (x, y, z) → (x, y, r_{1}z + r_{2}xy) with suitable r_{1}, r_{2}.

Proof of Lemma 9.3. If d = i + 1 then C ∈ Leg(ℝ^{3}) because in this case multigerm C is RL-equivalent to the multigerm (A_{2k}, l)_{i}^{∗} which is Legendrian with respect to the contact structure (dz − ydx).

Now we prove that if d > i + 1 then C ∉ Leg(ℝ^{3}). Assume, to get contradiction, that multigerm C is Legendrain with respect to contact structure (Adx+Bdy+Cdz). It is easy to see that in this case A(0) = B(0) = 0 and therefore we may assume that C ≡ 1. Then the function germs A(x, y, z) and B(x, y, z) satisfy the relations

2A(t^{2}, t^{2k+1}, 0) + (2k + 1)t^{2k−1}B(t^{2}, t^{2k+1}, 0) ≡ 0,  (9.4)
A(t, t^{i}, t^{d}) + it^{i−1}B(t, t^{i}, t^{d}) + dt^{d−1} ≡ 0.  (9.5)

Relation (9.4) implies

2\frac{∂A}{∂y}(0) + (2k + 1)\frac{∂B}{∂x} (0) = 0, \frac{∂A}{∂x}(0) = \frac{∂^{2}A}{∂x^{2}}(0) = ⋯ = \frac{∂^{2k−1}A}{∂x^{2k−1}}(0) = 0. (9.6)

Calculate the coefficient at t^{i} in (9.5). Using the second relation in (9.6) and the constrains 2 ≤ i ≤ 2k − 1, d > i + 1 we obtain

\frac{∂A}{∂y}(0) + i · \frac{∂B}{∂x} (0) = 0.  (9.7)

Relation (9.7) and the first relation in (9.6) imply \frac{∂A}{∂y}(0) = \frac{∂B}{∂x} (0) = 0. In this case the 1-form α = Adx + Bdy + dz is not contact: (α ∧ dα)(0) = 0.

10. Contact classification of multigerms consisting of 3 non-singular components

This section is devoted to contact classification of multigerms γ = (γ_{1}, γ_{2}, γ_{3}) consisting of 3 non-singular components. Notation. Given two germs A, B : (ℝ, 0) → (ℝ^{3}, 0) denote by ord(A, B) the order of tangency between A and B (if A and B are not tangent then ord(A, B) = 0).)

We restrict ourselves to the following singularity classes: T_{0,d}: ord(γ_{1}, γ_{2}) = 0, ord(γ_{1}, γ_{3}) = d ≥ 0 (up to permutation of the components); T_{1,1}: ord(γ_{1}, γ_{2}) = ord(γ_{1}, γ_{3}) = ord(γ_{2}, γ_{3}) = 1.

Proposition 10.1.

1. Let γ ∈ T_{0,d} ∪ T_{1,1}, d ≥ 0. Then γ ∈ Leg(ℝ^{3}) if and only if γ is planar.

2. Any multigerm γ ∈ T_{0,d} ∩ Leg(ℝ^{3}), d ≥ 0 is RL-equivalent to the multigerm

T_{0,d} : (t, 0, 0), (0, t, 0), (t, t^{d+1}, 0), d ≥ 0.

3. Any multigerm γ ∈ T_{1,1} ∩ Leg(ℝ^{3}) is RL-equivalent to the multigerm

T_{1,1} : (t, 0, 0), (t, t^{2}, 0), (t, bt^{2}, 0), b ∉ {0, 1}.

The parameter b in T_{1,1} is a modulus. Before proving Proposition 10.1 we present its corollary on Legendrian multigerms γ ∈T_{0,d}∪T_{1,1} in the contact space (ℝ^{3}, (dz− ydx)). Consider the following Legendrain multigerms in this contact space: T̃_{0,d} : γ_{1} : x = t, y = z = 0, γ_{2} : x = 0, y = t, z = 0,

γ_{3} : x = t, y = t^{d+1}, z = \frac{t^{d+2}}{d+2} ;

T̃_{1,1} : γ_{1} : x = t, y = z = 0, γ_{2} : x = t, y = t^{2}, z = \frac{t^{3}}{3} ,

γ_{3} : x = t, y = bt^{2}, z = \frac{bt^{3}}{3} , b ∉ \{0, 1\}.

These multigerms can be reduced to T_{0,d} and T_{1,1} by a change of coordinates z → z + kxy with a suitable k. Therefore Proposition 10.1 and Theorem 1.1 imply the following classification result.

Theorem 10.2. (real analytic and C^{∞} categories). Let γ be an Legendrain multigerm of the class T_{0,d} (respectively T_{1,1}) in the contact space (ℝ^{3}, (dz − ydx)). Then γ is contactomorphic to the multigerm T̃_{0,d} (respectively T̃_{1,1}).

Proof of Proposition 10.1. Consider at first the class T_{0,d}. Any multigerm γ ∈T_{0,d} is RL-equivalent to a multigerm of the form

(t, 0, 0), (0, t, 0), (t, t^{d+1}b(t), t^{d+1}c(t)), (b(0), c(0)) ≠ (0, 0).  (10.1)

Assume that c(0) = 0. Then b(0) ≠ 0. A change of coordinates (x, y, z) → (x, ky + xyf_{1}(x)z, z +xyf_{2}(x)) and a reparameterization t → k^{−1}t of the second component with suitable f_{1}(x), f_{2}(x) and k ≠ 0 reduces b(t) to 1 and c(t) to 0. We obtain the multigerm T_{0,d} ∈ Leg(ℝ^{3}).

Consider now the case c(0) ≠ 0. It is easy to prove that in this case multigerm (10.1) is not planar. On the other hand, it is quasi-homogeneous: a change of coordinates (x, y, z) → (x, y + zf_{1}(x), zf_{2}(x)) with suitable f_{1}(x), f_{2}(x), f_{2}(0) ≠ 0, reduces b(r) to 0 and c(r) to 1. By Theorem 1.3 (10.1) ∉ Leg(ℝ^{3}).

Now we will prove Proposition 10.1 for the class T_{1,1}. Any multigerm γ ∈T_{1,1} is RL-equivalent to a multigerm of the form

(t, 0, 0), (t, t^{2}, 0), (t, t^{2}b(t), t^{2}c(t)), (b(0), c(0)) ∉ (0, 0) ∪ (1, 0).  (10.2)

The condition (b(0), c(0)) ≠ (0, 0) (respectively (b(0), c(0)) ≠ (1, 0)) means that the order of tangency between the first and the third component (respectively between the second and the third component) is equal to 1.

Lemma 10.3. If c(0) ≠ 0 then (10.2) is RL-equivalent to the multigerm

(t, 0, 0), (t, t^{2}, 0), (t, 0, t^{2}).  (10.3)

If c(0) = 0 then (10.2) is RL-equivalent either to the multigerm T_{1,1} or to the multigerm of the form

(t, 0, 0), (t, t^{2}, 0), (t, bt^{2}, t^{3}), b ∉ {0, 1}.  (10.4)

It is easy to prove that multigerms (10.3) and (10.4) are quasi-homogeneous and not planar. Therefore Proposition 10.1 follows from Lemma 10.3 and Theorem 1.3.

Proof of Lemma 10.3. If c(0) ≠ 0 in (10.2) then b(t) can be reduced to 0 and c(t) to 1 by a change of coordinates of the form (x, y, z) → (x, y + zf_{1}(x), zf_{2}(x)) with suitable f_{1}(x), f_{2}(x), f_{2}(0) ≠ 0.

Assume now that in (10.2) one has c(0) = 0. Then b(0) ∉ {0, 1}. Consider the plane curve multigerm (t, 0), (t, t^{2}), (t, t^{2}b(t)). One can prove that under the assumption b(0) ∉ {0, 1} it is RL-equivalent to ((t, 0), (t, t^{2}), (t, bt^{2})), where b = b(0) is a modulus. Therefore (10.2) is RL-equivalent to a multigerm of the form

(t, 0, 0), (t, t^{2}, 0), (t, bt^{2}, t^{3}c̃(t)), b ∉ {0, 1}.

A change of coordinates (x, y, z) → (x, y, z + y(y − x^{i+1})f(x)) with a suitable f(x) reduces c̃(t) to the constant c̃ = c̃(0). If c̃ = 0 then we have the multigerm T_{1,1}. If c̃ ≠ 0 then c̃ can be reduced to 1 by scaling z and we obtain multigerm (10.4).

11. Appendix. Legendrian multigerms in the contact space (ℝ^{3}, (dz − ydx)): the projections to the (x, z) and the (x, y)-plane.

The contact classification of Legendrain singularities in Sections 6-10 is similar to the classification of multigerms of plane curves with respect to the whole group of diffeomorphisms. In the present section we we present two sufficient conditions for the contactomorphness of multigerms γ and γ̃ in the contact space (ℝ^{3}, (dz−ydx)) in terms of their projections to the (x, y) and the (x, z)-plane. These conditions are not necessary. We present examples showing that in general the contact classification of Legendrain multigerms cannot be reduced to the classification of multigerms of plane curves, even in the case of one component.

Denote by π_{x,z} and π_{x,y} the projections (x, y, z) → (x, z) and (x, y, z) → (x, y). Any Legendrain multigerm with components γ_{i} : x = a_{i}(t), y = b_{i}(t), z = c_{i}(t) in the contact space (ℝ^{3}, (dz − ydx)) is uniquely determined by the projection π_{x,y}(γ)

- a multigerm of a plane curve with components (a_{i}(t), b_{i}(t)). It is also uniquely determined by the projection π_{x,z}γ - a plane curve multigerm with components (a_{i}(t), c_{i}(t)) provided that the curves (a_{i}(t), b_{i}(t) have non-zero Taylor sries.

Notation. Denote by F the set of Legendrian multigerms in the contact space (dz − ydx) containing a component (a(t), b(t), c(t) such that the curve (a(t), c(t)) has the zero Taylor series.

Proposition 11.1. (real analytic and C^{∞} categories). Let γ and γ̃ be Legendrain multigerms in the contact space (ℝ^{3}, (dz−ydx)) beyond the set F. If the projections π_{x,z}(γ) and π_{x,z}(γ̃) are RL-equivalent then γ and γ̃ are contactomorphic.

Proof. Let (x, z) → (X(x, z), Z(x, z) be a local diffeomorphism sending π_{x,z}γ to π_{x,z}γ̃ up to reparameterization of the components. The condition that γ and γ̃ are Legendrain with respect to the 1-form dz − ydx and the assumption γ, γ̃ ∉ F imply that any component of the projections π_{x,z}γ and π_{x,z}γ̃ has up to reparameterization the form (±t^{r}, o(t^{r})) (r depends on the component). It follows that

\frac{∂X}{∂x} (0) ≠ 0, \frac{∂Z}{∂z} (0) = 0.  (11.1)

Consider the map

(x, y, z) → (X(x, z), Y (x, y, z), Z(x, z)), Y (x, y, z) = \frac{\frac{∂Z}{∂x} + y \frac{∂Z}{∂z}}{\frac{∂X}{∂x} + y \frac{∂X}{∂z}} .  (11.2)

Conditions (11.1) imply that (11.2) is a well-defined local diffeomorphism (ℝ^{3}, 0) → (ℝ^{3}, 0). One can check that (11.2) is a contactomorphism of the contact structure (dz −ydx). Since γ, γ̃ ∉ F then γ and γ̃ are uniquely determined by the projections π_{x,z}(γ) and π_{x}, z(γ̃) and it follows that contactomorphism (11.2) sends the image of γ to the image of γ̃. ^{7} □

On the other hand, if two Legendrain multigerms in the contact space (ℝ^{3}, (dz − ydx)) beyond the set F are contactomorphic then their projection to the (x, z)- plane do not need to be diffeomorphic.

Example 10.1. Consider the following Legendrain germs of singular curves in the contact space (ℝ^{3}, (dz − ydx)):

ψ : x = t^{3}, y = t^{4}, z = \frac{3t^{7}}{7} ,

ψ̃ : x = t^{3}, y = t^{4} + t^{5}, z = \frac{3t^{7}}{7} + \frac{3t^{8}}{8} .

The projections of the curves ψ and ψ̃ to the (x, z)-plane are diffeomorphic to (t^{3}, t^{7}) and (t^{3}, t^{7} + t^{8}) respectively. These two curves are not diffeomorphic. On the other hand, the curves ψ and ψ̃ are RL-equivalent to the same curve (t^{3}, t^{4}, 0) and by Theorem 1.1 they are contactomorphic.

7We should explain how the contactomorphism (11.2) is constructed. We use the formal notation y = dz/dx and write Y (x, y, z) = dZ/dX = \frac{(∂Z/∂x)dx+(∂Z/∂z)dz}{(∂Z/∂x)dx+(∂Z/∂z)dz} . We obtain (11.2) by dividing the nominator and the denominator by dx. To explain that these operations are legal one should consider the 3-dimensional manifold C^{3} of contact elements whose points are (p, l), where p is a point of the (x, z)-plane and l is a 1-dimensional subspace of T_{p}ℝ^{2}(x, z), see [3], [14]. The manifold C^{3} is a circle bundle over ℝ^{2}(x, z) endowed with a natural contact structure (α) defined as follows: a curve (p(t), l(t)) in C^{3} is tangent to (α) if ṗ(t) ∈ l(t) for all t. Any local diffeomorphism Φ of ℝ^{2}(x, z) can be lifted to a contactomorphism of the manifold (C^{3}, (α))

- the image under Ψ_{Φ} of a point (p, l) is the point ((Φ(p), Φ_{∗}l). The contactomorphism (11.2) is the germ of the lifting to C^{3} of the diffeomorphism (x, z) → (X, Z) at the point (p_{0}, l_{0}), where p_{0} = (0, 0), l_{0} = span\{ \frac{∂}{∂x} \}.

Now we consider the projection to the (x, y)-plane.

Proposition 11.2. (real analytic and C^{∞} categories). Let γ and γ̃ be Legendrain multigerms in the contact space (ℝ^{3}, (dz − ydx)). If the projections π_{x,y}(γ) and π_{x,y}(γ̃) are RL-equivalent via a local diffeomorphism preserving the volume form dx ∧ dy up to multiplication by a number then γ and γ̃ are contactomorphic.

Proof. Let (x, y) → (X(x, y), Y (x, y)) be a local diffeomorphism satisfying the assumption of Proposition 11.2. Then dX ∧ dY = rdx ∧ dy for some r ≠ 0 and consequently Y dX = rydx + dH(x, y) for some function germ H(x, y). It follows that the map Ψ : (x, y, z) → (X(x, y), Y (x, y), rz + H(x, y)) is a contactomorphism of the contact space (ℝ^{3}, dz − ydx)). Since γ and γ̃ are uniquely defined by their projections to the (x, y)-plane then Ψ sends the image of γ to the image of γ̃. □

Remark. One can show that a local diffeomorphism of the (x, y)-plane can be lifted to a local contactomorphism of the contact space (ℝ^{3}, dz − ydx)) if and only if it satisfies the assumption of Proposition 11.2.

In view of Proposition 11.2 we conclude this section with the following examples:

(a) there are contactomorphic Legendrain multigerms in the contact space (ℝ^{3}, (dz− ydx)) whose projections to the (x, y)-plane are not RL-equivalent;

(b) there are non-contactomorphic Legendrain multigerms in the contact space (ℝ^{3}, (dz − ydx)) whose projections to the (x, y)-plane are RL-equivalent. Example 10.2. Consider the germs ψ_{0}, ψ_{1}, ψ_{2} of Legendrain curve in the contact space (ℝ^{3}, (dz − ydx)) with the following projections to the (x, y)-plane: π_{x,y}ψ_{0} : x = t^{4}, y = t^{13} + t^{14} + t^{15}, π_{x,y}ψ_{1} : x = t^{4}, y = (t^{13} + t^{14} + t^{15}) · (1 + t^{4}),

π_{x,y}ψ_{2} : x = t^{4}, y = t^{13} + t^{14} + t^{15} + \frac{21t^{17}}{17} + \frac{22t^{18}}{17} + \frac{23t^{19}}{17} .

Let us show that the following statements hold:

1. The projections π_{x,y}ψ_{0} and π_{x,y}ψ_{1} are RL-equivalent;

2. The germs ψ_{0} and ψ_{1} are not RL-equivalent;

3. The germs ψ_{0} and ψ_{2} are RL-equivalent;

4. The projections π_{x,y}ψ_{0} and π_{x,y}ψ_{2} are not RL-equivalent.

Statement 1. is obvious - the diffeomorphism (x, y) → (x, y(1+x)) sends π_{x,y}(ψ_{0}) to π_{x,y}(ψ_{1}). Statement 3. follows from Proposition 11.1: the projections of ψ_{0} and ψ_{2} to the (x, z)-plane have the form

π_{x,z}(ψ_{0}) : x = t^{4}, z = f(t), π_{x,z}(ψ_{2}) : x = t^{4}, z = (1 + t^{4}) · f(t),

where f(t) = 4t^{17}/17 + 4t^{18}/18 + 4t^{19}/19. These projections are RL-equivalent, therefore the germs ψ_{0} and ψ_{1} are contactomorphic by Proposition 11.1.

The proofs of statements 2. and 4. are based on calculations. One can prove that the germ π_{x,y}ψ_{2} is RL-equivalent to a germ of the form

x = t^{4}, y = y = t^{13} + t^{14} + t^{15} + at^{19} + o(t^{19}), a ≠ 0,

and that a plane curve of this form is not RL-equivalent to the plane curve π_{x,y}ψ_{0}. One also can prove that the space curves ψ_{0} and ψ_{1} are RL-equivalent, respectively, to the curves of the form

x = t^{4}, y = t^{13} + t^{14} + t^{15}, z = t^{18} + at^{19} + o(t^{23}),
x = t^{4}, y = t^{13} + t^{14} + t^{15}, z = t^{18} + at^{19} + bt^{23} + o(t^{23}),

where a, b ≠ 0 and that these two curves are not RL-equivalent.

The singularities Example 10.2 are much more degenerate than in Example 10.1. This is in correspondence with results in [17] and Section 8. Denote by S the set of Legndrain germs in the contact space (ℝ^{3}, (dz − ydx)) such that the the contactomorphness of Legendrain germs γ, γ̃ beyond S is the same property as the diffeomorphness of their projections to the (x, y)-plane. Theorem 8.5 implies that codimS ≥ 6 in the space of all non-immersed Legendrain germs. Using Proposition 11.2 and results in [15] one can prove that codimS is substantially bigger than 6.

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Department of Mathematics, Technion, 32000 Haifa, Israel

   E-mail address: mzhi@techunix.technion.ac.il